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PUBLICATIONS  OF  SOWER,   POTTS  &.  CO.,  PHILADELPHIA. 


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IN  MEMORIAIA 
George  Davidson 


EDUCATION  DEPT. 


Professor  of  Geography 
University  of  California 


logical 


I 


The  uniform  testimony  of  teachers  who  have  introduced  these  grammars  is,  that  they  have  been 
most  agreeably  surprised  at  their  effects  upon  pupils.  They  are  easy  to  understand  by  the 
youngest  pupil,  and  the  lessons  before  dreaded  become  a  delight  to  teacher  and  pupils.  Ex- 
traordinary care  has  been  taken  in  grading  every  lesson,  modeling  rules  and  definitions  after 
a  definite  and  uniform  plan,  and  making  every  word  and  seatence  an  example  of  grammati- 
cal accuracy.    They  only  need  a  trial  to  supersede  all  others. 


PUBLICATIONS  OF  SOWER,   POTTS  &  CO.,   PHILADELPHIA. 


THE 

Normal  Series  of  Mathematics. 

BY   EDWARD   BROOKS,  A.M., 

PRINCIPAL   AND    PROFESSOR    OF    MATHEMATICS   IN    PENNSYLVANIA    STATE    N  SCHOOL 

AT    M1LLERSVILLE. 

This  series,  still  new  and  fresh,  has  had  an  extraordinary  success,  an  ?d  in 

very  many  of  the  best  Normal  Schools,  Academies,  Seminaries  and  Pu.  Is 

in  the  country.     Wherever  the  works  are  known  they  receive  the  highe* 
dations  of  leading  professors  and  teachers. 


PRICE 

Brooks's  Normal  Primary  Arithmetic  .  .  25  cts. 

The  Primary  contains  Mental  and  Written  Exercises  for  very  young  pupils.    Its  treatment  is 
very  plain,  easy  and  progressive. 


PRICE 

Brooks's  Normal  Elementary  Arithmetic  50  cts. 

The  Elementary  will  furnish  a  practical  business  education  in  a  shorter  time  and  with  less  labor 
than  any  other,  and  is  emphatically  the  work  for  those  pupils  who  must  be  qualified  for  com- 
mon business  in  one  or  two  terms. 


PRICE 

Brooks's  Normal  Mental  Arithmetic  ...  38  cts. 

The  Mental  is  a  philosophical  and  comprehensive  treatise  upon  the  Analysis  of  Numbers.  It  is 
easily  mastered  by  young  pupils,  and  those  who  accomplish  it  are  ready  to  grapple  with  the 
most  difficult  problems.  It  makes  logical  thinkers  on  all  subjects.  All  who  use  it  say  they 
cannot  be  induced  to  dispense  with  it. 


PRICE 

Brooks's  Normal  Written  Arithmetic  .  .  95  cts. 

The  Written  is  a  comprehensive  and  practical  work,  full  of  business  applications.  It  is  admirably 
arranged,  and  progresses  step  by  step  until  pupils  master  it  almost  without  conscious  effort. 
Its  treatment  is  novel  in  many  respects  and  very  interesting,  and  as  It  is  very  successful  in 
the  school-room,  it  is  very  popular  among  the  best  educators. 


PRICE 

Brooks's  Key  to  Mental  Arithmetic  ....  8  .38 
Brooks's  Key  to  Elementary  Arithmetic  .  .50 
Brooks's  Key  to  Written  Arithmetic  ...    1.00 

In  addition  to  the  matter  usually  contained  in  Keys,  the  first  and  last  of  these  contain  many  valu- 
able suggestions  on  the  best  methods  of  teaching  Arithmetic. 


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ELEMENTARY  ARITHMETIC: 


EMBRACING 


A  COURSE  OF  EASY  AND  PROGRESSIVE  EXERCISES  IN 
ELEMENTARY  WRITTEN  ARITHMETIC; 


DESIGNED  FOR 


PRIMARY  SCHOOLS,  AND  PRIMARY  CLASSES  IN  COMMON  SCHOOLS, 
GRADED  SCHOOLS,  MODEL  SCHOOLS,  ETC. 


BY 


EDWARD  BROOIvS,  A.  M. 

FmiNCIPAJ.  AND   PROFESSOR  OP   MATHEMATICS  IN   PENNSYLVANIA   STATE   NORMAL  SCHOOL,   AND    AUTHOR  O* 

NORMAL   PRIMARY   ARITHMETIC,  NORMAL    MENTAL    ARITHMETIC,  NORMAL   WRITTEN 

ARITHMETIC,  ELEMENTARY   GEOMETRY,  ETC. 


\ 


\  1  >  v  V 


"  The  highest  science  is  the  greatest  simplicity." 


PHILADELPHIA: 
SOWER,   BARNES  &  POTTS, 

530  MARKET  ST.  and  523  MINOR  ST. 


. 


Office  of  the  Controllers  of  Public  Schools,  ^ 

First  District  of  Pennsylvania.  V 

Philadelphia,  March  29, 1869.     ) 

At  a  meeting  of  the  Controllers  of  Public  Schools,  First  District  of 
Pennsylvania,  held  at  the  Controllers'  Chamber,  Tuesday,  March  9, 
186S,  rfie  following  resolution  was  adopted  : — 

Resolved,  That  the  Normal  Series  of  Arithmetics,  comprising  "Brooks's 
Nonucii  Primary  Arithmetic,"  "  Brooks's  Normal  Elementary  Arithme 
tic."  '■  Brooks's  Normal  Mental  Arithmetic,"  and  "Brooks's  Normal 
Writtp-i  Arithmetic,"  by  Edward   Brooks,  Esq.,  be  and   the   same  are 
hereby  nlaced  on  the  list  of  books  to  be  used  in  the  Public  Schools  of 

this  District. 

From  the  minutes : 

H.  W.  Halliwell,  Sec'y. 


Entered,  according  to  Act  of  Congress,  in  the  year  1865,  by 

EDWARD  BROOKS, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  in  and  for  the  Eastern 

District  of  Pennsylvania. 


ME*:iS  &  DUSENBERY,  ELECTROTYPERS.  SHERMAN  &  CO.,  PRINTERS. 


The  Board  of  School  Trustees  for  the  State  of  Maryland,  recom- 
mend Brooks's  Normal  Arithmetics  and  Fewsmith's  Grammars,  for  use  in 
all  the  Public  Schools  of  that  state. 


The  Board  of  School  Commissioners  for  the  City  of  Baltimore. 
hove  adopted  Brooks's  Normal  Series  and  Fewsmith's  Grammars,  fo! 
exclusive  use  in  the  Public  Schools  of  that  city. 


- 


PREFACE. 


The  object  of  this  work  is  to  furnish  young  pupils 
with  an  introductory  course  of  Written  Arithmetic, 
Realizing  the  necessity  of  such  a  course  in  connec- 
tion with  Mental  Exercises,  the  author  gave  quite  a 
large  collection  of  written  exercises  in  his  Primary 
Mental  Arithmetic.  So  popular  was  this  feature 
that  he  was  soon  urged  either  to  increase  the  amount 
of  such  matter  or  prepare  another  work  treating 
exclusively  of  the  elements  of  Written  Arithmetic. 
Believing  that  the  latter  plan  would  be  the  more 
acceptable,  and  give  more  completeness  to  the  arith- 
metical course,  the  present  volume  has  been  pre- 
pared. 

This  work  will  be  found  to  possess  at  least  five 
distinguishing  features :  1st,  Simplicity ;  2d,  Gra- 
dation ;  3d,  Practical  Character  of  the  Problems ; 
4th,  Variety  of  Problems ;  5th,  Educational  Cha- 
racter. 

Simplicity. — Great  care  has  been  taken  to  make 
the  definitions,  explanations,  solutions,  rules,  etc. 
so  simple  that  the  youngest  pupils  can  easily  under- 
stand them.  In  doing  this,  however,  the  scientific 
character  of  the  subject  has  not  been  sacrificed;  for 
it  should  ever  be  remembered  that  the  highest  science 
is  the  greatest  simplicity ;  and,  conversely,  the  greatest 
simplicity  is  the  highest  science. 

3 

m£89£98 


PREFACE. 


Gradation. — The  gradation  of  the  work  will  be 
found  one  of  its  most  distinctive  and  valuable  fea- 
tures. A  frequent  criticism  upon  elementary  works 
is  their  lack  of  gradation,  their  sudden  transitions 
from  the  easy  to  the  difficult,  from  the  simple  to  the 
complex.  To  avoid  this  fault,  great  pains  have  been 
taken,  and,  it  is  believed,  with  success.  For  example, 
see  the  exercises  in  Addition,  Subtraction,  Multi- 
plication, and  Division,  where  the  problems  are  ar- 
ranged into  classes  and  cases  with  respect  to  their 
length  and  difficulty.  The  same  spirit  of  gradation 
will  be  found  running  through  the  whole  work. 

Practical  Problems. — Arithmetics  have  been 
criticized  for  the  abstract  and  unpractical  character 
of  their  problems.  To  avoid  this  error,  I  have  given 
a  large  number  of  practical  problems,  drawn  from 
the  actual  events  of  life.  Among  these  are  His- 
torical, Geographical,  and  Biographical  problems  ; 
problems  on  the  battles  of  the  Revolution;  farm- 
ers', merchants',  etc.,  problems.  Such  problems  will 
not  only  add  interest  to  the  study  of  arithmetic,  but 
present  much  valuable  information  to  the  pupil. 

Variety  of  Problems. — "Variety  is  the  spice  of 
life,"  in  the  school-room  as  well  as  out  of  it.  It  is 
a  great  mistake  to  keep  pupils  upon  Addition  for 
several  months,  until  they  have  thoroughly  mastered 
it,  then  upon  Subtraction  for  a  corresponding  length 
of  time,  and  so  on  for  the  other  fundamental  opera- 
tions. The  better  way  is  to  give  them  a  fair  know- 
ledge of  Addition,  then  take  them  to  Subtraction, 
and,  after  they  are  somewhat  familiar  with  this, 
give  them  exercises  combining  Addition  and  Sub- 
traction, and  thus  through  the  fundamental  rules, 


PREFACE. 


leaving  each  subject  before  the  pupil  wearies  of 
it,  and  returning  to  it  again  and  again,  until  it  is 
thoroughly  mastered.  In  this  manner  tiresome 
monotony  is  avoided,  and  the  task  of  the  learner 
rendered  interesting  and  attractive.  This  is  a  fea- 
ture of  the  present  work  which  it  is  believed  will 
commend  it  to  intelligent  instructors. 

Educational  Character.  —  This  work,  like  the 
others  of  the  same  series,  is  characterized  by  an 
educational  spirit.  It  is  not  a  mere  collection  of  pro- 
blems and  rules  for  the  training  of  a  pupil  to  labor 
like  a  machine.  The  spirit  of  analysis  runs  through 
it,  making  it  normal  in  the  broadest  sense  of  the 
term.  Its  object  is  to  teach  pupils  to  think  as  well 
as  to  work  'problems, — to  develop  mind  as  well  as  the 
power  of  computation. 

Cherishing  the  hope  that  it  may  aid  teachers  in 

their  arduous  labors,  and  become  a  favorite  with  the 

little  girls  and  boys  of  our  common  schools,  I  now 

intrust  it  to  the  decision  of  a  kind  and  appreciative 

public. 

EDWARD  BROOKS. 

Statb  Normal  School, 

May  20,  1865. 


1* 


SUGGESTIONS  TO  TEACHERS. 

1.  This  book  is  designed  to  be  put  into  the  hands  of  young 
pupils  soon  after  they  begin  the  study  of  Primary  Mental 
Arithmetic.  When  pupils  can  add  and  subtract  orally  with 
some  facility,  they  will  be  prepared  for  this  work. 

2.  The  Introduction  is  not  designed  to  be  studied  by  the  pupils, 
but  indicates  a  course  of  Oral  Instruction  in  the  elements  of 
arithmetic ;  and  it  is  suggested  that  these  exercises  receive  the 
attention  which  their  importance  demands.  The  pupils  should 
have  careful  drill  upon  these  before  beginning  the  study  of  a 
written  arithmetic;  and  such  exercises  will  be  found  valuable 
during  the  entire  course,  especially  upon  commencing  a  new 
subject. 

3.  Pupils  should  solve  the  problems  upon  the  slate  at  their 
seats,  and  also  be  required  to  work  them  out  upon  the  black- 
board, and  explain  them.  In  assigning  problems  at  the  board, 
the  same  problem  may  be  given  to  the  whole  class,  or  each  pupil 
may  receive  a  different  problem.  Sometimes  one  method  is  pre- 
ferred, and  sometimes  the  other.  The  object  should  be  thorough- 
ness and  accuracy,  and  at  the  same  time  variety  and  interest. 

4.  In  many  cases  two  forms  of  explanation  have  been  given; 
one  a  full  logical  form,  the  other  an  abridged,  mechanical  one. 
The  object  of  the  first  is  to  present  the  reasoning  process  in  full ; 
the  object  of  the  second,  to  give  the  steps  in  the  method  of  opera- 
tion. Though  it  is  important  that  the  pupil  should  understand 
the  complete  logical  form,  yet  for  the  ordinary  recitation,  with 
young  pupils,  the  abbreviated  form  may  be  preferred.  It  will 
economize  time,  and  better  secure  that  which  is  mainly  aimed  at 
in  primary  written  arithmetic, — facility  and  accuracy  of  operation. 

5.  Where  a  pupil  has  difficulty  with  a  problem  owing  to  its 
being  a  little  complicated,  let  the  teacher  lead  him  from  one  step 
to  another,  and  so  on  to  the  end,  by  a  judicious  series  of  questions, 
leading  him  to  analyze  the  problem,  and  thus  unfold  its  com- 
plexity. This  will  be  much  better  for  the  pupil  than  to  pick  up 
his  slate  and  work  the  problem  for  him.  By  attention  to  these 
suggestions,  and  to  such  other  points  as  will  occur  to  the  mind 
of  the  intelligent  teacher,  it  is  hoped  that  the  progress  of  the  pupil 
will  be  rapid  and  thorough. 

6 


INTRODUCTION. 


OEAL   EXEECISES. 

The  following  remarks  and  oral  exercises  are  designed  to  illus- 
trate the  manner  in  which  the  elementary  principles  of  numbers 
should  be  presented  to  the  young  pupil. 

LESSON   I. 

NAMING    NUMBERS. 

Since  our  first  ideas  of  numbers  are  derived  from  visible  objects, 
the  child's  first  lessons  in  the  science  of  numbers  should  be  given 
with  such  objects.  These  objects  may  consist  of  apples,  nuts,  books, 
pencils,  grains  of  corn,  or  any  thing  the  teacher  may  find  con- 
venient. The  German  schools  for  young  children  are  generally 
supplied  with  small  cubical  blocks  to  be  used  in  the  first  lessons 
on  numbers.  The  arithmetical  frame  is  the  most  convenient  for  gene- 
ral practice. 

Counting. — The  names  of  numbers  are  acquired  simultaneously 
with  the  idea  of  numbers.  Both  of  these  are  given  in  the  process  of 
counting.  By  counting  we  do  not  mean  merely  speaking  the  words 
one,  two,  three,  etc.,  but  that  these  words  should  be  used  in  connec- 
tion with  objects,  so  that  the  pupil  may  know  what  the  words  mean. 
I  have  known  pupils  who  could  run  off  these  words  very  glibly, 
even  as  far  as  fifty,  without  any  definite  idea  of  their  meaning. 
Hence  the  pupil's  first  lesson  in  numbers  should  be  in  counting. 
The  method  suggested  is  as  follows: — 

The  teacher,  standing  before  the  class,  holding  some  object,  as  a 
book,  in  his  hand,  says,  "What  do  I  hold  in  my  hand?"  Pupils: 
"A  book."  Teacher:  "  How  many  books?"  Pupils:  "One  book." 
The  teacher,  taking  up  another  book,  says,  "How  many  books  in 
my  hand   now?"     Pupils:    "Two   books."     And   so   on,    until    the 

7 


8  INTRODUCTION. 

pupils  can  number  any  collection  of  objects  which  the  teacher  holds 
in  his  hand. 

After  this  introduction,  he  will  take  the  arithmetical  frame  and  con- 
tinue the  exercises  upon  it.  These  exercises  may  be  made  lively  by 
increasing  or  diminishing  the  number  by  several  at  the  same  time. 
Little  counting  games,  with  grains  of  corn,  will  also  be  found  very 
interesting  to  young  pupils. 

Let  exercises  of  this  kind  be  continued  until  the  class  can  count 
well.  If  the  pupils  can  count  when  they  enter  school,  this  exercise 
need  not  be  continued  long. 

LESSON   II. 

ADDITION    AND    SUBTRACTION. 

After  the  pupils  can  readily  number  a  collection  of  objects, — that 
is,  have  the  idea  of  numbers  and  the  names  of  numbers, — they  should 
be  taught  to  unite  and  separate  them.  The  next  thing  in  order,  there- 
fore, is  addition  and  subtraction.  Instruction  in  these  processes 
should  be  given  in  accordance  with  the  following  principles. 

1.  The  first  lessons  in  addition  and  subtraction  should  be  given  with 
visible  objects.  This  principle  is  founded  upon  the  law  of  mental 
development,  and  is  so  evident  that  it  need  not  be  urged.  Indeed, 
if  the  teacher  neglects  it,  the  pupil  will  adopt  it  himself,  by  adding 
with  his  fingers,  etc. 

2.  Addition  and  subtraction  should  be  taught  together  in  primary  oral 
instruction.  This  is  evident,  since  the  two  ideas  are  logically  related. 
Thus,  as  soon  as  the  pupil  learns  that  2  and  3  are  5,  he  sees  that  5 
diminished  by  2  equals  3,  or  5  diminished  by  3  is  2.  Convenience 
also  dictates  the  same  method.  In  the  primary  schools  of  Germany, 
the  two  processes  are  combined,  in  the  manner  illustrated. 

Exercise. — The  order  of  exercises  will  now  be  given.  The  pupils 
should  first  increase  and  diminish  by  one,  as  far  as  12,  then  by  two, 
then  by  three,  etc.     The  exercise  would  be  somewhat  as  follows: — 

Teacher  takes  one  book  in  his  hand,  and  asks,  "How  many 
books  have  I  in  my  hand?" 

Pupils  answer,  "One  book." 

Teacher,  putting  another  book  in  his  hand,  asks,  "How  many 
books  have  I  now  ?" 

Pupils  answer,  "Two  books." 

Teacher:   "How  many,  then,  are  one  book  and  one  book?" 

Pupils:   "Two  books." 

Teacher     "How  many  books  have  I  in  my  hand  now?" 


INTRODUCTION.  9 

Pupils:   "Two  books." 

Teacher:  "I  will  take  one  book  away;  now  how  many  books 
remain  ?' 

Pupils:   "One  book." 

Teacher:  "One  book  taken  from  two  books,  then,  leaves  how 
many  books?" 

Pupils:   "One  Dook.'- 

Let  the  teacher  now  take  the  arithmetical  frame,  and  proceed  in  the 
same  way,  increasing  2,  3,  4,  etc.,  up  to  12  with  one,  and  diminish- 
ing 3,  4,  5,  6,  etc.,  up  to  13  with  one,  each  time  reversing  the  addi- 
tion. Then  take  one  and  increase  it  by  two,  obtaining  three ;  then 
reverse  the  process  and  diminish  three  by  tico,  and  so  on  until  12  is 
increased  by  two,  and  14  diminished  by  two.  Proceed  in  the  same 
way  with  3,  4,  etc.,  until  the  pupils  can  add  and  subtract  by  ones, 
twos,  threes,  etc.,  up  to  twelves. 

LESSON   III. 
practical  exercises. 

The  following  exercises  will  be  found  valuable  in  teaching  pupils 
to  add  and  subtract  with  readiness  and  accuracy.  Frequent  drill 
upon  such  exercises  is  recommended. 

First. — A  valuable  exercise  to  give  the  pupils  readiness  in  adding 
and  subtracting  is  the  following:  Let  the  teacher  name  two  num- 
bers, and  require  the  pupils  to  give  first  their  sum  and  then  their 
difference:   thus,  teacher  says,  "5  and  2." 

Pupils:  "5  and  2  are  7,  and  2  from  5  leaves  3."  After  a  little 
practice  they  may  omit  naming  the  numbers,  and  merely  say,  "the 
sum  is  7,  the  difference  is  3." 

To  vary  this,  the  boys  may  give  the  sum,  and  the  girls  the  differ- 
ence, and  vice  versa;  or,  if  the  class  is  all  of  one  sex,  a  division  may 
be  made,  one  part  giving  the  sum,  and  the  other  part  the  difference. 
In  this  and  the  following  exercises  care  should  be  taken  that  small 
numbers  be  used  at  first,  until  the  pupils  attain  the  ability  to  use 
larger  numbers  with  ease  and  readiness 

Second. — Another  valuable  exercise  consists  in  the  teacher  select- 
ing some  number,  and  then  giving  one  part  of  this  number  himself, 
requiring  the  pupils  to  give  the  other  part.  For  example,  suppose 
8  to  be  the  number  selected:  the  teacher  says,  "five,"  pupils  answer, 
"three,"  teacher,  "tivo,"  pupils,  "six"  etc.  etc. 

Third. — The  following  exercise  will  also  be  found  valuable  in  im- 
parting the  art  of  adding  and  subtracting  with  readiness  and  accu- 


one. 
two 
three 
four 
five 
six 
seven 
eight 
nine 


10  INTRODUCTION. 

racy.     Let  the  teacher  write  the  words  one,  two,  three,  etc.,  on  the 

board,  forming  two  columns  as  indicated  in  the  mar-       ( 

ein      Call  the  first  column  additive,  and  the  second 

6  •  ,    ,,  •  n    one 

subtractive.     The  teacher  then  with  the  pointer  will 

indicate  the  number,  the  operation  being  indicated 

tilTCC 

by  the  column.     When  he  points  to  figures  in  the  first 

column,    the    number    which    they   indicate    will   be  J 

added,  but  when  he  points  to  any  figure  in  the  second,  J  . 

the  number  indicated  will  be  subtracted  from  the  re- 

.        .  ,    .       ,     seven 
suit    which    the    pupils    have    previously    obtained. 

After  the  Arabic  characters  have  been  given,  instead 

of  writing  the  words  in  columns,  the  figures  may  be 

employed  for  the  same  purpose. 

Fourth. — The  pupils  should  also  be  required  to  add  by  twos,  threes, 
etc.,  merely  naming  the  results,  as  follows;  2,  4,  6,  8,  etc.,  3,  6,  9, 
etc.,  until  the  additions  can  be  readily  given. 

It  will  be  well  to  commence  also  with  one,  and  count  by  twos, 
thus:  1,  3,  5,  7,  etc.  ;  also  commence  at  1,  and  count  by  threes, 
thus:  1,  4,  7,  10,  etc.;  also  at  2,  thus:  2,  5,  8,  11,  etc.,  continuing 
the  addition  as  far  as  it  may  be  thought  desirable. 

Let  the  pupil  be  exercised  in  a  similar  manner  in  adding  by  fours, 
fives,  etc.,  up  to  twelves.  Such  exercises  should  be  continued  day 
after  day,  in  connection  with  the  lessons  which  precede  and  follow 
this  lesson,  until  great  facility  is  acquired  in  the  operations. 

These  exercises  may  be  conducted  sometimes  in  concert  and  some- 
times singly.  While  one  is  adding  alone,  let  the  others  keep  careful 
watch  for  errors  ;  a  good  degree  of  interest  may  thus  be  created, 
each  pupil  trying  to  obtain  the  largest  sum  before  making  a  mis- 
take. 

It  is  evident  that  the  teacher  can  give  great  variety  to  these  exer- 
cises ;  and  the  author  suggests  that  they  will  be  found  of  very  great 
utility. 

LESSON   IV. 

WRITING    NUMBERS. 

The  pupil  should  now  be  taught  how  to  write  numbers.  In  fact, 
the  writing  of  numbers  should  be  introduced  very  soon  after  the 
naming  of  numbers.  The  following  exercises  should,  therefore,  be 
combined  with  the  exercises  of  Lesson  III.     The  method  suggested 

is  as  follows  :  — 

The  teacher,  standing  at  the  board,  with  some  objects,  as  two  books 
in  his  hand,  inquires,  "What  do  I  hold  in  my  hand?" 


INTRODUCTION.  11 

Answer : — "  Books." 

Teacher  :    "  How  many  books  ?" 

Pupils  :    "  Two  books." 

Teacher  (writing  two  books  upon  the  board)  asks,   "What  hava 
I  written  upon  the  board?" 

Pupils:   "Two  books." 

Teacher  :   "Are  there  two  books  on  the  board  ?" 

Pupils  :   "Yes,  sir." 

Teacher:  "If  there  are  two  books  on  the  board,  then  what  are 
these  I  hold  in  my  hand?" 

Pupils  :    "Why,  those  are  two  books  also." 

Teacher:  "Well,  if  that  is  two  books  on  the  board,  and  these 
are  two  books  in  my  hand,  then  they  must  both  be  the  same." 

Pupils:  "Oh,  no,  that  on  the  board  is  the  icords  two  books,  but 
what  you  have  in  your  hand  are  the  things  two  books." 

Thus  the  important  distinction  between  the  thing  and  the  expression 
of  it  is  attained. 

The  teacher  will  now  send  the  pupils  to  the  board,  and  let  them 
write  the  words  one,  two,  etc,  and  have  them  solve  problems  in 
addition  and  subtraction  with  them.  If  the  pupils  cannot  write 
(and  I  presume  that  will  generally  be  the  case  at  the  time  such  exer- 
cises are  appropriate),  the  teacher  can  illustrate  by  performing  the 
written  exercises  himself. 

The  pupils  will  soon  see  the  great  labor  of  employing  the  written 
words,  and  will  realize  the  necessity  of  an  arithmetical  written  lan- 
guage different  from  that  which  is  used  in  ordinary  writing.  They 
are  now  prepared  for  the  Arabic  characters;  and  the  manner  in 
which  these  should  be  introduced  will  now  be  given. 

Characters. — The  pupils  being  now  prepared  for  the  Arabic 
characters,  let  the  teacher  give  first*  in  their  order  the  nine  digits. 
They  should  be  exercised  in  naming  and  writing  these  until  they 
are  familiar  with  them  and  can  make  them  with  considerable  ease 
and  neatness.  They  should  then  be  required  to  solve  problems  with 
them  in  addition  and  subtraction,  the  teacher  giving  no  problem 
at  present  which  involves  a  number  greater  than  nine. 

Combinations. — When  the  class  are  familiar  with  these  characters, 
they  are  to  be  taught  to  combine  them  to  express  the  larger  num- 
bers. There  are  two  methods  of  doing  this,  quite  different  in 
principle  and  form.     We  present  both. 

First  Method. — By  this  method  we  give  the  combined  characters, 
without  explaining  the  principle  of  the  combination.  Thus,  wo 
teach  that  10  represents  ten,  11,  eleven,  12,  twelve,  etc.,  without  any 


VI  INTRODUCTION. 

reference  to  tens  and  units.  This  method  is  not  quite  so  philoso- 
phical as  the  second  method,  but  will  be  found  preferable  in  practice 
with  young  learners  in  oral  instruction.  When  pupils  study  written 
arithmetic  from  the  book,  I  would  use  the  other  method. 

We  would  give  these  expressions  as  far  as  twenty,  and  then  drill 
the  pupils  in  reading  and  writing  them  until  they  are  quite  familiar 
with  them.  We  would  next  give  the  expressions  from  twenty  to 
thirty,  and  drill  in  like  manner,  and  thus  continue  as  far  as  one 
hundred. 

After  the  pupils  are  familiar  with  this  method  of  writing  numbers 
as  far  as  100,  the  teacher  may  then  show  them  the  principle  of  the 
combination,  that  the  figure  in  the  first  place  represents  units,  in 
the  second  place  tens,  etc.  When  this  is  understood,  we  would  re- 
quire the  class  to  analyze  these  expressions,  as  follows: — 

Problem. — Analyze  25  [twenty-five). 

A?ialysis. — In  25,  the  5  represents  5  units,  and  the  2  represents  2 
tens. 

The  class  should  also  be  drilled  upon  questions  like  the  follow- 
ing:— How  many  units  in  2  tens?  In  3  tens?  etc.  How  many  tens 
in  20  (twenty)  1  etc.  How  many  units  and  tens  in  24  (twenty-four)  1 
They  should  also  be  required  to  solve  little  problems  in  addition 
and  subtraction  with  these  characters. 

Second  Method. — The  other  method  commences  by  explaining  the 
principle  of  the  combination;  that  is,  that  10  represents  1  ten;  11,  1 
ten  and  1  unit;  12,  1  ten  and  2  units,  etc.,  afterward  showing  that  11 
(1  ten  and  1  unit)  is  the  same  as  eleven,  etc. 

This  may  be  done  by  making  ten  marks  upon  the  board,  and 
then  commencing  a  second  row  with  one  mark ;  these  will  be  repre- 
sented by  11  (1  ten  and  1  unit)  ;  then  have  the  pupils  count  them, 
and  they  will  see  that  11  (1  ten  and  1  unit)  stands  for  eleven.  The 
same  may  be  done  with  12,  13,  etc. 

The  pupil  should  be  drilled  in  reading  and  writing  numbers,  until 
he  is  entirely  familiar  with  the  subject.  Haste  here  is  "bad 
speed."  A  thorough  knowledge  of  Notation  and  Numeration  wil) 
dispel  the  usual  difficulties  of  Addition,  Subtraction,  Multiplication, 
ind  Division. 

LESSON   V. 

MULTIPLICATION    AND    DIVISION. 

After  the  pupil  has  become  quite  familiar  with  the  elementary 
processes  of  Addition  and  Subtraction,  he  is  prepared  to  take  up 
Multiplication  and  Division.    The  first  instruction  in  these  processes 


INTRODUCTION.  13 

should  be  given  by  oral  exercises,   and   in  accordance  with  the 
following  principles: 

1.  Multiplication  should  be  presented  as  a  special  case  of  Addition 
Thus,  the  pupil  should  be  taught  that  two  2's  are  4,  since  2  -j-  2  ==4, 
or  that  three  times  2  are  6,  since  2  taken  three  times,  or  2  -j-  2  — |-  2, 
equals  6.  and  so  on  for  the  other  products.     The  pupil  will  then 
understand  the  nature  of  the  subject. 

2.  Division  should  be  taught  as  reverse  multiplication.  Thus,  it  should 
be  shown  that  6  contains  3  two  times,  since  tivo  times  3  are  6,  and  so 
on  for  other  quotients.  In  this  way  the  quotients  are  immediately 
derived  from  the  products.  Division  may  be  taught  as  concise  sub- 
traction,  but  the  process  of  reverse  multiplication  is  more  convenient 
and  simple.  "When  thus  taught,  the  pupil  will  not  need  to  commit 
a  distinct  Division  Table. 

3.  The  pupil  should  be  taught  to  construct  the  Multiplication  Table. 
The  pupil  should  not  be  required  to  commit  a  Multiplication  Table 
without  knowing  how  it  was  obtained,  or  the  use  of  it.  He  should 
first  be  taught  to  derive  the  products  for  himself,  by  addition,  and 
then  be  required  to  commit  them,  to  avoid  the  labor  of  obtaining 
them  every  time  he  wishes  to  use  them.  In  this  way  he  will  study 
them  with  more  interest,  and  learn  them  with  greater  ease. 

4.  Multiplication  and  Division  should  bu  taught  simultaneously,  or  at 
least  very  nearly  so.  As  soon  as  the  pupil  learns  that  2  times  3  are  6, 
he  is  able  to  see  that  6  equals  two  3's,  or  that  6  contains  3  two  times; 
and  the  same  is  true  for  the  other  products.  Division,  therefore, 
should  not  be  deferred  until  the  whole  Multiplication  Table  is 
learned,  as  has  generally  been  the  practice,  but  should  be  early 
introduced  and  studied  in  connection  with  Multiplication. 

We  now  present  the  following  exercise,  which  is  designed  to  sug- 
gest the  manner  in  which  the  first  principles  of  multiplication  and 
division  may  be  taught. 

EXERCISE. 

Teacher  (making  two  marks  on  the  board,  as  is  indi- 
|   |         cated  in  the  margin)  asks,  "How  many  marks  have  I 
made?" 

Pupils:   "Two  marks. 

Teacher  (making  two  marks  under  the  former  two,  aa 
is  indicated  in  the  margin,  inquires) :   "How  many  times 
J   j         have  I  made  two  marks?" 
Pupils:    "Two  times." 

2 


14  INTRODUCTION. 

Teacher  :   "  How  many  marks  are  there  in  all  ?" 

Pupils  :   "  Four  marks." 

Teacher:   "  How  many,  then,  are  two  times  two  marks?" 

Pupils  :   "  Two  times  two  marks  are  four  marks." 

Teacher  (leaving  the  four  marks  upon  the  board,  asks):  "How 
many  marks  are  there  on  the  board?" 

Pupils  .   "  Four  marks." 

Teacher:   "Are  they  arranged  in  twos,  or  threes?" 

Pupils  :  "In  twos." 

Teacher:  "  How  many  twos  are  there  in  these  four  marks?'* 

Pupils  :   "  Two  twos." 

Teacher:   "Four,  then,  contains  two  how  many  times?" 

Pupils  :  "Two  times." 

The  teacher  will  now  write  six  marks  upon  the  board, 

as  in  the  margin,  and  ask,   "  How  many  times  have  I 

|   |   |       written  three  marks?"     "How  many  are  there  in  all?" 

"How  many,  then,  are  two  times  three?"     "Are  these 

6  marks  arranged  in  twos,  threes,  or  fours?"     "How  many  times 

three  marks  are  there  ?"     "  How  many  threes,  then,  are  there  in 

six?"     "Six,  then,  contains  3  how  many  times?" 

These,  or  similar  exercises,  should  be  continued  up  to  two  times 
12,  each  time  reversing  the  process,  and  obtaining  a  quotient.  Then 
proceed  in  the  same  way  with  three  times,  four  times,  etc.,  on  to 
twelve  times.  In  this  manner  the  pupils  may  be  led  to  obtain,  and 
then  commit,  the  products  and  quotients,  usually  given  in  tables, 
which  are  always  learned  with  so  much  hesitation  and  hard  study. 

In  practice,  it  will  be  well  to  obtain  all  the  products  of  "two  times," 
before  deriving  the  quotients.  Questions  similar  to  those  in  the  Pri- 
mary Arithmetic,  p.  82,  may  also  be  given.  The  pupil  should  also 
be  taught  to  write  the  table  of  products  upon  the  slate  or  board, 
thus: 

1X2  =  2  4X2=   8 

2X2  =  4  5X2  =  10 

8X2  =  6  etc. 

Pupils  generally  have  considerable  difficulty  in  committing  the 
Multiplication  Table;  the  teacher  can  lessen  the  labor  in  several 
ways.  1st.  By  having  the  pupils  make  it  for  themselves,  and  write 
it  on  the  slate  or  blackboard.  2d.  By  conoert  recitation.  3d.  By 
singing  the  table  to  some  appropriate  tune.  4th.  Reciting  by  the  old 
method  of  "going  up,"  or  "trapping." 

To  make  pupils  rapid  and  accurate  in  the  mechanical  processes  of 


INTRODUCTION.  15 

addition,  subtraction,  multiplication,  and  division,  the  following  ex- 
ercise is  practised  by  some  teachers,  with  excellent  results.  Let  the 
teacher  write  four   columns  of   figures  on 

the    blackboard,   as  is   represented  in  the       (-]-)    ( — )    (X)    (-*-) 
margin,  the  first  column  being  additive,  the         1111 
next  subtractive,  etc.,  as  is  indicated  by  the  2         2         2         2 

symbols  placed  above  them.  The  teacher,  3  3  3  3 
with  the  pointer,  will  point  out  certain  4  4  4  4 
figures,  the  corresponding  numbers  being  5  5  5  5 
added,  subtracted,  multiplied,  or  divided,  6  6  6  6 
as  is  indicated  by  the  symbol  at  the  head  of  7  7  7  7 
the  column.  Care,  of  course,  must  be  taken  8  8  8  8 
not  to  require  a  division  by  a  number  that  9  9  9  9 
is   not   exactly   contained.      This   exercise 

may  be  continued  for  many  recitations,  in  connection  with  the  follow 
ing  lessons,  with  great  advantage  to  the  pupils. 

LESSON  VI. 

TERMS    AND    PRINCIPLES. 

The  following  exercises  are  designed  to  suggest  the  manner  of 
giving  the  terms  of  the  fundamental  rules,  and  also  of  deriving  some 
of  the  principles  of  each. 

ADDITION    AND    SUBTRACTION. 

Teacher:   "What  have  I  in  my  hand?" 

PuriLS  :   "  Two  books." 

Teacher.  "What  is  the  difference  between  the  two  and  the 
books?" 

Pupils:  "The  books  are  the  things,  and  the  two  tells  how  many 
things." 

Teacher:  "The  two  denotes  the  number  of  books.  What,  then, 
is  a  number?" 

Pupils  :    "Why,  it  is  the  how  many  of  any  thing." 

Teacher  :  "Very  well .  remember,  also,  that  a  single  thing,  or  one 
of  a  collection,  is  called  a  unit." 

Teacher:    "When  I  say  two  apples,  what  2  do  I  mean?" 

Pupils:    "Two  apples." 

Teacher:   "When  I  say  tiro,  what  do  I  mean?" 

Pupils  :   "  We  do  not  know.'' 

Teacher:    "What  2  may  I  mean?  any  two?" 

Pupils:   "Yes,  sir;  any  two  you  choose." 

Teacher:    "You  see  a  difference,   then,   between   two  and    ttn 


16  INTRODUCTION. 

books.  Very  well ;  I  will  give  the  name  which  denotes  this  difference. 
When  I  say  2,  3,  etc.,  without  telling  what  2  or  3,  it  is  called  an 
abstract  number;  but  when  I  give  the  name  of  the  objects  with  the 
number,  it  is  called  a  concrete  number." 

Tell  which  of  the  following  numbers  are  abstract,  and  which  con^ 
crete : — 

2  cows — three — four — 4  books — 7  hens — eight — 5 — 4 — 10  pigs — 
8  geese — 7 — 6 — 11 — 14  horses. 

Teacher:   How  many  are  3  and  5? 

Teacher:  When  we  unite  two  numbers  into  one,  in  this  way,  the 
result  is  called  the  sum,  and  the  process  is  called  addition. 

Teacher:  What  is  the  sum  of  2  and  3?  4  and  6?  7  and  8? 

Teacher  :  What  is  the  sum  of  3  cows  and  5  turnips  ? 

Teacher  :  Why  can  you  not  add  them  ? 

Teacher:  If  they  were  all  the  same,  could  you  add  them  ? 

Teacher:  Numbers  which  express  the  same  kind  of  objects  are 
similar  concrete  numbers,  and  those  which  denote  different  objects 
are  dissimilar  concrete  numbers. 

Teacher  :  What  kind  of  numbers  can  be  added,  then,  and  what 
kind  cannot  be  added  ? 

How  many  remain  when  we  take  3  apples  from  5  apples  ? 

The  process  of  taking  one  number  from  another  is  called  sub- 
traction. 

The  number  which  is  taken  away  is  called  the  subtrahend,  the 
number  from  which  it  is  taken  is  called  the  minuend,  and  the  result 
is  called  the  difference,  or  remainder. 

If  you  subtract  4  from  9,  which  is  the  minuend,  which  the  subtra- 
hend, and  which  the  remainder? 

If  you  add  the  remainder  and  subtrahend  together,  will  it  pro- 
duce the  minuend  ? 

If  you  subtract  the  difference  from  the  minuend,  what  will  it 
equal? 

Can  you  subtract  3  apples  from  5  potatoes  ? 

Why  can  you  not  subtract  them  ? 

Are  these  similar  or  dissimilar  concrete  numbers? 

If  they  were  similar,  could  they  be  subtracted  ? 

What  kind  of  numbers,  then,  can  be  subtracted,  and  what  kind 
eannot  ? 

PRINCIPLES    OF    MULTIPLICATION    AND    DIVISION. 

1.  When  we  find  the  result  of  a  number  taken  any  number  of 
times,  the  process  is  called  multiplication. 


INTRODUCTION.  17 

2.  The  number  taken  a  certain  number  of  \imes  is  called  the 
multiplicand. 

3.  The  number  which  denotes  how  many  times  the  multiplicand 
is  taken  is  called  the  multiplier. 

4.  The  result  obtained  is  called  the  product.    Each  of  these  three 
is  called  a  term. 

5.  What  is  the  product  of  8  apples  multiplied  by  4? 

6.  In  this  problem,  which  is  the  multiplicand,  which  the  multi- 
plier, which  the  product? 

7.  When  we  take  8  apples  4  times,  is  the  result  apples,  or  some- 
thing else  ? 

8.  Can  the  product  be  any  thing  else  than  apples? 

9.  The  product,  then,  is  of  the  same  denomination  as  what  term? 

10.  Can  we  take  8  apples  4 peaches  times,  or  simply  4  times? 

11.  Is  4  an  abstract  or  a  concrete  number  ?  What  kind  of  a  num- 
ber, then,  must  the  multiplier  be  ? 

12.  When  we  find  how  many  times  one  number  is  contained  in 
another,  the  process  is  called  division. 

13.  The  number  which  contains  the  other  is  called  the  dividend, 
the  number  contained  is  called  the  divisor,  and  the  number  denoting 
how  many  times  the  divisor  is  contained  is  called  the  quotient. 

14.  If  we  divide  8  apples  by  2  apples,  is  the  result  apples?  If 
not,  what  is  it? 

15.  Are  2  apples  contained  in  8  apples  4  peaches  times,  or  4 
apples  times,  or  simply  4  times? 

1G.  What  kind  of  a  number  is  4,  and  what  kind  of  a  number, 
then,  must  the  quotient  always  be  ? 

17.  How  many  times  2  equal  8  apples  ?  Is  2,  or  2  pears,  contained 
any  number  of  times  in  8  apples  ? 

18.  What  2  are  contained  any  number  of  times  in  8  apples? 

19.  The  divisor,  then,  must  be  of  the  same  denomination  as  what 
term? 

LESSON   VII. 

TABLE  OF  FUNDAMENTAL  RULES. 

We  now  present  the  tables  of  the  four  fundamental  rules,  for  such 
teachers  as  wish  to  use  them.  The  author  suggests  that  the  ele- 
mentary sums  and  differences  are  better-  taught  by  the  exercises 
which  have  been  already  suggested  than  by  the  study  of  these 
tables.  The  Multiplication  Table,  however,  must  be  thoroughly 
committed,  and  then  the  elementary  quotients  may  be  derived  from 
these  products,  or  by  the  study  of  the  Division  Table. 

2* 


18 


INTRODUCTION. 


ADDITION   TABLE. 


2  and 

3  and 

4  and 

5  and 

0  are  2 

0  are  3 

0  are  4 

0  are  5 

1  "   3 

1  "   4 

1  «   5 

1  "     6 

2  "   4 

2  «   5 

2  "   6 

2  «  7 

3  "   5 

3  "   6 

3  »   7 

3  "   8 

4  "   6 

4  "   7 

4  S*   8 

4  "   9 

5  "   7 

5  "   8 

5  «   9 

5  "  10 

G  "   8 

6  "   9 

6  "  10 

6  "  11 

7  "   9 

7  "  10 

7  "  11 

7  "  12 

8  »  10 

8  "  11 

8  "  12 

8  "  13 

9  "  11 

9  "  12 

9  "  13 

9  "  14 

10  "  12 

10  "  13 

10  "  14 

10  "  15 

11  "  13 

11  "  14 

11  "  15 

11  "  16 

12  "  14 

12  "  15 

12  "  16 

12  »  17 

6  and 

7  and 

8  and 

9  and 

0  are  6 

0  are  7 

0-  are  8 

0  are  9 

1  "  7 

1  "  8 

1  "   9 

1  "  10 

2  "   8 

2  "   9 

2  "    10 

2  "  11 

3  "   9 

3  "  10 

3  »  11 

3  »  12 

4  "  10 

4  "  11 

4  «  -12- 

4  "  13 

5  "  11 

5  "  12 

5  "  13 

5  "  14 

6  "  12 

6  "  13 

6  "  14 

6  »  15 

7  "  13 

7  "  14 

7  "  15 

7  "  16 

8  "  14 

8  "  15 

8  "  16 

8  "  17 

9  "  15 

9  "  16 

9  "  17 

9  »  18 

10  »  16 

10  "  17 

10  "  18 

10  "    19 

11  "  17 

11  "  18 

11  «  19 

11  "  20 

12  "  18 

12  "  19 

12  "  20 

12  "    21 

10  and 

11  and 

12  and 

13  and 

0  are  10 

0  are  11 

0  are  12 

0  are  13 

1  "  11 

1  "  12 

1  »  13 

1  "  14 

2  "  12 

2  "  13 

2  "  14 

2  "  15 

3  "  13 

3  "  14 

3  «  15 

3  "  16 

4  "  14 

4  «  15 

4  »«  16 

4  "    17 

5  "  15 

5  "  16 

5  «  17 

5  "  18 

6  "  16 

6  "  17 

6  «  18 

G  "  19 

7  «  17 

7  "  18 

7  "  19 

7  "  20 

8  "  18 

8  "  19 

8  "  20 

8  "  21 

9  "  19 

9  "  -20 

9  "  21 

9  <l  22 

10  "  20 

10  "  21 

10  "  22 

10  »  23 

11  "  21 

11  "  22 

11  »  23 

11  "  24 

12  »  22 

12  "  23 

12  "  24 

12  "  25 

INTRODUCTION. 


19 


SUBTRACTION   TABLE. 


1  " 

1  from 

2  from 

3  from 

■ 

4  from 

1  leaves  0 

2 

leaves  0 

3 

leaves  0 

4 

leaves  0 

2   « 

1 

3 

1 

4 

1 

5 

1 

3   * 

2 

4 

2 

5 

2 

6 

"    2 

4   < 

3 

5 

3 

6 

3 

7 

3 

6   < 

4 

6 

4 

7 

4 

8 

4 

6   < 

5 

7 

5 

8 

5 

9 

5 

7   < 

1    6 

8 

6 

9 

6 

10 

6 

8   « 

4    < 

9 

7 

10 

7 

11 

7 

9   < 

8 

10 

8 

11 

8 

12 

8 

10   < 

9 

11 

9 

12 

9 

13 

9 

11   ♦ 

<   10 

12 

»   10 

13 

"   10 

14 

"   10 

12   < 

<   11 

13 

"   11 

14 

"   11 

15 

«      11 

13   < 

<   12 

14 

"   12 

15 

"   12 

16 

"   12 

51 

rom 

6  from 

7  from 

8  from 

5  lea 

.ves  0 

6 

leaves  0 

7 

leaves  0 

8 

leaves  0 

6   « 

♦    1 

7 

1 

8 

1 

9 

1 

7   < 

2 

8 

2 

9 

2 

10 

"    2 

8   « 

3 

9 

3 

10 

3 

11 

3 

9   « 

4 

10 

4 

11 

4 

12 

4 

10   < 

5 

11 

5 

12 

5 

13 

5 

11   < 

6 

12 

6 

13 

6 

14 

6 

12   < 

7 

13 

7 

14 

7 

15 

7 

13   < 

8 

14 

8 

15 

8 

16 

8 

14   « 

■    9 

15 

9 

16 

9 

17 

9 

15   < 

<   10 

16 

"   10 

17 

"   10 

18 

"   10 

16   « 

1   11 

17 

"   11 

18 

"   11 

19 

"   11 

17   « 

1   12 

18 

"   12 

19 

it      ^o 

20 

"   12 

91 

rom 

10  from 

11  from 

12  from 

9  lea 

ves  0 

10 

leaves  0 

11 

leaves  0 

12 

leaves  0 

10   « 

1 

11 

"    1 

12 

1 

13 

"    1 

11   < 

2 

12 

2 

13 

"    2 

14 

2 

12   < 

3 

13 

3 

14 

3 

15 

3 

13   < 

4 

14 

4 

15 

4 

16 

4 

14   « 

5 

15 

5 

16 

5 

17 

5 

15   ■ 

6 

16 

6 

17 

6 

18 

"    6 

16   • 

17 

7 

18 

7 

19 

7 

17   « 

8 

18 

8 

19 

8 

20 

8 

18   ' 

9 

19 

9 

20 

9 

21 

9 

19   « 

*      10 

20 

"   10 

21 

"   10 

"   10 

20   ■ 

■   11 

21 

"   11 

22 

"   11 

23 

"   11 

21 

1   12 

22 

"   12 

23 

"  12 

24 

"   12 

20 


INTRODUCTION. 


MULTIPLICATION   TABLE. 


Once 

2  times 

3  times 

■■■■■-     ■      —    -    n 

4  times 

1      is       1 

1   are     2 

1    are     3 

1    are    4 

2      "       2 

2     "       4 

2     «      6 

2     "      8 

3      "       3 

3     "      6 

3     "      9 

3     "    12 

4      "       4 

4     "      8 

4     "    12 

4     "    16 

5      "       5 

5     "    10 

5     "     15 

5     "    20 

6      "       6 

6     «    12 

6     "    18 

6     "    24 

7      "       7 

7     "    14 

7     "    21 

7     "    28 

8      "       8 

8     "     16 

8     "    24 

8     "    32 

9      "       9 

9     "    18 

9     "    27 

9     "    36 

10      "     10 

10     "    20 

10     "    30 

10     «    40 

11      "     11 

11     "    22 

11     '*    33 

11     "    44 

12      "     12 

12     «    24 

12     "    36 

12     "    48 

5  times 

6  times 

7  times 

8  times 

1   are     5 

1   are     6 

1   are     7 

1   are     8 

2     "     10 

2     "     12 

2     "     14 

2     u     16 

3     "     15 

3     "     18 

3     "    21 

3     "    24 

4     "    20 

4     "    24 

4     "    28 

4     "    32 

5     "    25 

5     «    30 

6     "    35 

5     "    40 

6     "    30 

6     "    36 

6     "    42 

6     "    48 

7     "    35 

7     "    42 

7     "    49 

7     "    56 

8     "    40 

8     "    48 

8     "    56 

8     "    64 

9     "    45 

9     "    54 

9     "    63 

9     "    72 

10     "    50 

10     "    60 

10     il    70 

10     "    80 

11     "    65 

11     »    66 

11     "    77 

11     "    88 

12     "    60 

12     "    72 

12     "    84 

12     "    96 

9  times 

10  times 

11  times 

12  times 

1   are     9 

1  are    10 

1  are    11 

1  are    12 

2     "     18 

2   "      20 

2   "      22 

2  "      24 

3     "    27 

3   «      30 

3   "      33 

3  "      36 

4     "    36 

4   «      40 

4   "      44 

4  "      48 

5     "    45 

5   "      50 

5   "      55 

5  "      60 

6     "    54 

6   "      60 

6   »      66 

6  "      72 

7     "    63 

7    "      70 

7   "      77 

7   "      84 

8     "    72 

8    "      80 

8   «      88 

8  "      96 

9     «    81 

9    "      90 

9   "      99 

9  "    108 

10     "    90 

10   "    100 

io  «  no 

10  "    120 

11     «    99 

li  "  no 

11    "    121 

11  "    132 

12     "  108 

12   »    120 

12   "    132 

12  "    144 

INTRODUCTION. 


21 


DIVISION   TABLE. 


lin 

2  in 

3  in 

4  in 

1        1  time 

2       1  time 

3       1  time 

4       1  time 

2       2  times 

4       2  times 

6       2  times 

8       2  times 

3       3" 

6       3      " 

9       3" 

12       3      " 

4       4" 

8       4" 

12       4      " 

16       4      " 

5       5" 

10       5      " 

15       5      " 

20       5      " 

6       6" 

12       6      " 

18       6      " 

24       6      " 

7       7" 

14       7      " 

21       7      " 

28      7      " 

8       8" 

16       8      " 

24      8      " 

32       8      " 

9       9" 

18       9      " 

27       9      " 

36       9      " 

10     10      " 

20     10      " 

30     10      " 

40     10      " 

11     11      " 

22     11      " 

33     11      " 

44     11      " 

12     12      " 

24     12      " 

36     12      " 

48     12      " 

5  in 

6  in 

7  in 

8  in 

5       1  time 

6       1  time 

7       1  time 

8       1  time 

10       2  times 

12       2  times 

14       2  times 

16       2  times 

15       3      " 

18       3      " 

21       3      " 

24       3      " 

20       4      " 

24       4      " 

28       4      " 

32       4      " 

25       5      " 

30       5      " 

35       5      " 

40       5      " 

30       6      " 

36       6      " 

42       6      " 

48       6      " 

35       7      " 

42       7      " 

49       7      " 

56       7      " 

40       8      " 

48       8      " 

56       8      " 

64       8      " 

45       9      " 

54       9      " 

63       9      " 

72       9      " 

50     10      " 

60     10      " 

70     10      " 

80     10      " 

55     11      " 

66     11      " 

77     11      " 

88     11      " 

GO     12      " 

72     12      " 

84     12      " 

96     12      " 

9  in 

10  in 

11  in 

12  in 

9     1  time 

10     1  time 

11     1  time 

12     1  time 

18     2  times 

20     2  times 

22     2  times 

24     2  times 

27     3     " 

30     3     " 

33     3     " 

36     3     " 

36     4     " 

40     4     " 

■14     4     " 

48     4     " 

43     5     " 

50     5     " 

56    5     " 

60     5     " 

54     6     " 

00     6     " 

66     6     " 

72     6     " 

63     7     " 

70     7     " 

77     7     " 

84     7     " 

72     8     " 

8     " 

88     8     " 

96     8     " 

81     9     " 

90     9     " 

99    9     " 

108     9     " 

90  10     " 

100  10     " 

110  10     " 

120  10     " 

11      " 

no  n    " 

121   11     " 

132  11     " 

108   i  !     •• 

120  12     " 

132  12     " 

144  12     " 

THE 

NORMAL 


ELEMENTARY  ARITHMETIC. 


SECTION  I. 

1.  A  Unit  is  a  single  thing,  as  a  book,  pen,  apple. 

2.  A  Number  is  one  or  more  things,  as  two  books, 

three  pens. 

3.  Arithmetic  is  the  science  of  numbers  and  the  art 

of  using  them. 

4.  Mental  Arithmetic   is  the    solving  of  problems 
without  the  aid  of  written  characters. 

5.  Written  Arithmetic  is   the    solving  of  problems 
with  written  characters. 

6.  In   studying   arithmetic  we  first  learn  to  name 
numbers,  and  then  learn  to  write  them. 

NUMEKATIOK 

7.  Numeration  is  the  art  of  naming  numbers,  and 
of  reading  them  when  they  are  written. 

8.  We  will  give  the  names  of  some  of  the  numbers 
up  <o  one  hundred. 

six  ; 


one  ; 
two ; 
three; 
four ; 
five ; 


seven; 
eight; 
nine; 
ten; 

23 


24 


NOTATION. 


eleven,  or  one  and  ten  ; 
twelve,  or  two  and  ten ; 
thirteen,  or  three  and  ten ; 
fourteen,  or  four  and  ten  ; 
fifteen,  or  five  and  ten ; 
sixteen,  or  six  and  ten ; 
seventeen,  or  seven  and  ten  ; 
eighteen,  or  eight  and  ten  ; 
nineteen,  or  nine  and  ten ; 
twenty,  or  two  tens; 
twenty-one,  or  two  tens  and  one; 


twenty-two,  or  two  tens  and  two ; 
thirty,  or  three  tens; 
thirty-one,  or  three  tens  and  one; 
forty,  or  four  tens  ; 
forty-one,  or  four  tens  and  one ; 
fifty,  or  five  tens  ; 
fifty-one,  or  five  tens  and  one ; 
sixty,  or  six  tens  ; 
sixty-one,  or  six  tens  and  one ; 
seventy,  or  seven  tens  ; 
etc.,  etc. 


Note. — The  pupil  should  be  drilled  upon  these  equivalent  forms 
of  naming  numbers,  as  a  preparation  for  Notation.  The  teacher  or 
pupil  may  fill  out  the  omissions. 

ABABIC  NOTATION. 
9.  Notation  is  the  art  of  writing  numbers. 
lO.  Figures. — In  writing  numbers  we  use  the  follow- 
ing ten  characters,  called  figures. 


1  expresses 

one. 

6 

expresses 

six. 

2         « 

two. 

7 

u 

seven. 

3 

three. 

8 

« 

eight. 

4 

four. 

9 

u 

nine. 

5 

five. 

0 

a 

naught 

11.  Combination. — By  these  figures  and  their  combi- 
nations all  numbers  can  be  expressed. 

The  method  of  combining  them  is  as  follows : — 

1st.  A  figure  standing  alone  expresses  units  or  ones. 

2d.  When  two  figures  are  together,  the  one  in  the  first 
place  at  the  right  expresses  units,  the  one  in  the  second 
place  expresses  tens. 

3d.  A  figure  in  the  third  place  expresses  hundreds,  in 
the  fourth  place  thousands,  etc. 

1*2.  Thus,  in  25,  the  2  expresses  2  tens,  and  the  0 
expresses  5  units.  We  will  illustrate  this  by  the  follow- 
ing table. 


NOTATION. 


25 


10  is  one  ten. 


20 
30 
40 
50 
60 
70 
80 
90 
100 


two  tens, 
three  tens, 
four  tens, 
five  tens, 
six  tens, 
seven  tens, 
eight  tens, 
nine  tens, 
one  hundred. 


11  is  1  ten  and  one. 


12 
23 

34 
47 

58 
65 
79 

86 
105 


1  ten  and  two. 

2  tens  and  three. 

3  tens  and  four. 

4  tens  and  seven. 

5  tens  and  eight. 

6  tens  and  five. 

7  tens  and  nine. 

8  tens  and  six. 

1  hundred  and  five. 


13.  The  pupils  will  now  write  and  read  the  follow- 
ing numbers : 


13 
15 
17 
19 
21 


24 
26 

27 
29 
30 


33 

36 
38 
39 
40 


41 
43 
45 

50 
52 


57 
60 
68 
63 

70 


74 
76 
82 
95 
126 


14.  The  pupils  will  now  learn  the  names  of  the  first 
twelve  places,  as  represented  in  the  following 


NUMERATION    TABLE. 


CO 


Names. 


Places'. 
Periods. 


03 

a 
o 


© 


03 

S3 
c3 

o 

•** 

03 

© 

— 

-*- 

• 
93 

■ 
03 

a 

1 

03 

o 

od 

o 
la 

03 

© 

© 

T3" 

T. 

a 
o 

1 

© 

u 

•73 

o 

r3 

a 

33 

© 

QQ 

© 

u 

T3 

■/' 

a 

I— I 

a 
M 

a 

r— I 

a 

d 

a 

© 

© 

3 

© 

J3 
H 

© 

4 

4 

4 

4 

4 

4 

4 

4 

4 

1 

4 

P=T 

A 

-*- 

© 

^3 

^a 

.a 

^ 

^a 

.a 

M 

pi 

+A 

f— H 

-^-> 

-*-* 

h_> 

-^j 

T3 

^3 

03 

I— 1 

i—i 

Ci 

CO 

i- 

o 

>-o 

•5 

CO 

<N 

4th. 


8J. 


2d. 


1st. 


15.  The  pupil  should  now  be  drilled  upon  questions 
similar  to  the  following. 

3 


26  NOTATION. 


Required  the  names  of  the  following  places  : 


1.  First. 

2.  Third. 

3.  Fourth. 


4.  Second. 

5.  Fifth. 

6.  Sixth. 


7.  Eighth. 

8.  Seventh. 

9.  Ninth. 


Required  the  places  of  the  following  : 


1.  Tens. 

2.  Hundreds. 

3.  Ten-thousands. 


4.  Thousands. 

5.  Millions. 

6.  Hundred-thousands. 


16.  Periods, — For  convenience  in  writing  and  read- 
ing numbers,  we  arrange  the  figures  in  periods  of  three 
places  each,  as  shown  by  the  table. 

The  first  three  places  make  the  first  or  units,  period, 
the  second  three  places  make  the  second  or  thousands, 
-period,  etc. 

Required  the  period  and  place  of  the  following : 


1.  Hundreds. 

2.  Thousands. 

3.  Millions. 

4.  Ten -thousands. 


5.  Tens. 

6.  Ten-millions. 

7.  Hundred-thousands. 

8.  Hundred-millions. 


17.  The  combination  of  figures  to  express  a  number 
forms  a  numerical  word.  Thus,  25  is  the  numerical  word 
which  means  the  same  as  twenty-five.  These  numerical 
words  may  be  analyzed. 

PROBLEMS. 

1.  Analyze  the  numerical  word  324. 

Analysis. — The  4  represents  four  units,  the  2  represents  two  tens, 
the  3  represents  three  hundreds :  hence  the  numerical  word  is  thrct 
hundred  and  twenty-four. 

Analyze  the  following  : 

2.  426  4.  652  6.  853  8.  395  10.  1234  12.  43762 

3.  357  5.  785  7.  687  9.  785  11.  5678  13.  85967 


NOTATION. 


27 


IS.  "We  will  now  give  some  exercises  in  Numeration 
and  Notation. 


RULE    FOR    NUMERATION. 

I.  Begin  at  the  right  hand,  and  separate  the  numerical 
word  into  periods  of  three  figures  each. 

II.  Then  begin  at  the  left  hand,  and  read  each  period  as 
if  it  stood  alone,  giving  the  name  of  each  period  except  the 
last. 

1.  What  number  is  expressed  by  3254789  ? 


Solution. — We  separate  the  numerical  word  into 
periods  of  three  figures  each,  as  in  the  margin.  The 
third  period  is  3  millions,  the  second  period  is  254 
thousands,  the  first  is  789 ;  hence  the  number  is  3 
millions,  254  thousands,  789. 


Eead  the  following 


OPERATION. 

3,254,789 


2. 

2384 

6. 

64327 

10. 

6321456 

3. 

7428 

7. 

52105 

11. 

78535217 

4. 

6321 

8. 

43246 

12. 

852106721 

5. 

8357 

9. 

785625 

13. 

12345678935 

19.  Having  learned  to  read  numerical  words,  the 
pupils  are  now  prepared  to  write  them.  From  the 
principles  we  have  given  we  derive  the  following 


RULE   FOR   NOTATION. 

Begin  at  the  left,  and  write  each  figure  in  order  towards 
the  right,  giving  each  figure  its  proper  place,  and  filling 
the  vacant  places  with  ciphers. 

1.  Write  the  number  four  thousand  three  hundred 
and- seven. 

Solution. — We  write  the  4  in  thousands'  place,  the     operation. 
3  in  hundreds'  place,  the  7  in  units'  place,  and,  since  4307 

there  arc  no  tens,  we  write  a  naught  in  tens'  place. 


28  NOTATION. 

Write  the  following  in  figures : 

2.  Three  thousand  and  seventy-five.  Ans.  3075. 

3.  Five  thousand  six  hundred  and  fifty. 

4.  Seven  thousand  eight  hundred  and  four. 

5.  Twenty-three  thousand  four  hundred  and  ninety. 

6.  Twenty-five  thousand  three  hundred  and  seven. 

7.  Two  hundred  and  six  thousand  four  hundred  and 
six. 

8.  Four  hundred  and  eighty-six  thousand  nine  hun- 
dred and  eight. 

9.  Seven  hundred  and  forty-three  thousand  four  hun- 
dred and  ninety. 

10.  Two  millions,  three  hundred  thousand  four  hun- 
dred and  eighty. 

11.  Four  millions,  five  hundred  and  six  thousand  and 
twenty-five. 

12.  Six  billions,  six  millions,  six  thousands,  six  hun- 
dred and  six. 

Remark. — Pupils  should  be  drilled  in  exercises  like  those  given, 
until  they  can  read  and  write  numbers  readily. 


ROMAN  NOTATION. 

The  Eoman  Method  of  Notation  employs  seven  let- 
ters of  the  Eoman  alphabet.  Thus,  I  represents  one ; 
V,  five;  X,  ten;  L,  fifty ;  C,  one  hundred;  D,  five  hun* 
dred ;  M,  one  thousand. 

To  express  other  numbers  these  characters  are  com- 
bined according  to  the  following  principles  : — 

1.  Every  time  a  letter  is  repeated  its  value  is  repeated. 

2.  When  a  letter  is  placed  before  one  of  greater  value,  the 
difference  of  their  values  is  the  number  represented. 

3.  When  a  letter  is  placed  after  one  of  a  greater  value, 
the  sum  of  their  values  is  the  number  represented. 


NOTATION. 


29 


4.  A  dash  'placed  over  a  letter  increases  its  value  a  thou- 
%and  fold.     Thus,  VII  denotes  seven  thousand. 


ROMAN    TABLE. 


I 

II 

III 

IV 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XIV 

XV 

XIX 

XX 


One. 

Two. 

Three. 

Four. 

Five. 

Six. 

Seven. 

Eight. 

Nine. 

Ten. 

Eleven. 

Fourteen. 

Fifteen. 

Nineteen. 

Twenty 


XXX 

XL 

L 

LX 

LXX 

XC 

c 
cc 

D 

DC 

DCCCC 

M 

MM 

MCLX 


Thirty. 

Forty. 

Fifty. 

Sixty. 

Seventy. 

Ninety. 

One  hundred. 

Two  hundred. 

Five  hundred. 

Six  hundred. 

Nine  hundred. 

One  thousand. 

Two  thousand. 

One  thousand  one  hun' 


MDCCCLIX1859.  [dred  and  sixty 


The  Eoman  Method  is  named  after  the  Romans,  who 
invented  and  used  it..  It  is  now  employed  to  denote 
the  chapters  and  sections  of  books,  pages  of  preface  and 
introduction,  and  in  other  places  for  prominence  and 
distinction. 

LUMBERMEN'S  NOTATION. 

Lumbermen  in  marking  lumber  employ  a  modifica- 
tion of  the  Roman  Method  of  Notation.  The  first  four 
characters  are  like  the  Roman ;  the  others  are  as 
follows ; 


A  Al  All  AIM  X  X  XI 

5  6  7  8            9  10  11 

A  Al  All  AW  XIX  X  X\    M 

15  16  17  18           19  20  21 


XII 

12 


22 


80 


90 


28 

m 

100 


///TtTTi 
200 


XIII 

13 

XIII 
23 


^\  M  m   JK\III  XMIII  Ifk  X    W^     "K 

25         26  2"  28  29         30      40  60  fin 


60 


14 

w 

24 
70 


30  ADDITION. 

SECTION    II. 
ADDITION. 

20.  Addition  is  the  process  of  finding  the  sum  of 
two  or  more  numbers. 

21.  The  Sum  is  a  number  which  contains  as  many 
units  as  the  numbers  added. 

22.  The  sign  of  Addition  is  -j->  and  is  read  plus. 
The  sign  of  Equality  is  =  ,  and  is  read  equals,  or  equal 
to.     Thus,  4  -j-  5  =  9,  is  read  4  plus  5  equals  9. 

Case  I. 

23.  To  add  when  the  sum  of*  a  column  is  not  more 
than  nine  of  that  column. 

24.   CLASS  I. —Problems  of  one  column. 

1.  What  is  the  sum  of  2,  3,  and  4  ? 

OPERATION. 

Solution  — We  write  the  numbers  one  under  an-  2 

itther  and  commence  at  the  bottom  to  add.     4  and  3  3 

are  7  and  2  are  9.     Hence  the  sum  is  nine.  4 


9 


EXAMPLES   FOR   PRACTICE. 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

(8.) 

2 

4 

6 

1 

7 

3 

1 

1 

1 

0 

5 

0 

4 

2 

3 

3 

3 

3 

2 

2 

6 

(9.)   (10.)  (11.)  (12.)  (13.)  (14.)  (15.)  (16.)  (17.) 
324623553 
512014212 
143262234 


ADDITION.  31 

18.  What  is  the  sum  of  2,  0,  3,  4? 

19.  What  is  the  sum  of  3,  1,  0,  2,  3? 

20.  What  is  the  sum  of  2,  2,  3,  0,  1  ? 

21.  What  is  the  sum  of  4,  1,  0,  2,  1? 

22.  What  is  the  sum  of  3,  0,  2,  0,  1,  3  ? 

25.    CLASS  II.— Problems   of   more    than   one 
column. 

1.  What  is  the  sum  of  21,  32,  43  ? 


Solution. — We  write  the  numbers  so  that  units 
stand  under  units,  and  tens  under  tens,  and  com- 
mence at  the  right  to  add.  The  sum  of  the  units  is 
3  and  2  are  5  and  1  are  6,  which  we  write  in  units' 
place.  The  sum  of  the  tens  is  4  and  3  are  7  and  2 
are  9,  which  we  write  in  tens'  place.  Hence  the  sum 
is  96. 


OPERATION. 
21 
32 

43 

96 


EXAMPLES 

FOR 

PRACTICE. 

(2-) 

(3.) 

(*•) 

(5.) 

(6.) 

31 

20 

34 

12 

15 

23 

14 

20 

23 

40 

24 

25 

15 

54 

34 

78 

(7.) 

(8.) 

(9.) 

(10.) 

(11.) 

121 

214 

610 

234 

361 

213 

312 

156 

432 

215 

432 

153 

213 

123 

123 

766 


(12.) 

(13.) 

(14.) 

(15.) 

(16.) 

612 

314 

712 

416 

201 

105 

212 

150 

141 

305 

271 

271 

137 

222 

281 

# 


32 


(17.) 

2021 

18.) 
5234 

(19.) 
6141 

(20.) 
7124 

(21.) 
6214 

3514 

1321 

1213 

1321 

322 

2361 

2141 

2032 

2042 

1211 

(22.) 
34123 

(23.) 
41210 

(24.) 
50273 

(25.) 
1234 

(26.) 
23071 

14310 

13025 

17202 

4012 

12303 

20341 

21613 

21310 

3701 

20413 

11111 

12030 

10101 

1020 

21210 

Kequired  the  sum 

27.  Of  2031,  1234,  3122,  and  1010. 

28.  Of  1207,  3040,  2430,  and  2112. 

29.  Of  2051,  3027,  1500,  and  1320. 

30.  Of  21021,  2712,  12032,  102,  and  21. 
hi.   Of  12201,  23021,  2142,  and  12012. 

Case  II. 

26.  To  add  when  the  sum  of*  any  column  is  more 
than  nine. 

27.  CLASS  I.— Problems  of  one  column. 

1.  What  is  the  sum  of  7,  6,  and  8  ? 


Solution. — We  write  the  numbers  one  under 
the  other,  and  commence  at  the  bottom  to  add.  8 
and  6  are  14  and  7  are  21.  We  place  the  1 
under  the  column,  and  the  2  in  tens'  place. 


OPERATION. 

7 
6 
8 

21  Ans. 


13 


EXAMPLES   FOR   PRACTICE. 


(2.) 

(3.) 

(4-) 

(5.) 

(6.) 

(7-] 

3 

7 

8 

6 

8 

3 

4 

2 

2 

3 

2 

9 

6 

5 

7 

7 

6 

7 

ADDITION. 


33 


(8.) 

5 
4 
3 
2 


(9.) 

7 
3 
8 
5 


(10.) 

6 
3 

8 

7 


(ll.) 
4 
3 
6 
5 


(12.) 

7 
3 
8 
5 


(13.) 
6 
7 
3 
8 


(14.) 

(15.) 

(16.) 

(17.) 

(18.) 

(19.) 

8 

3 

7 

2 

4 

7 

7 

7 

1 

0 

5 

8 

3 

2 

3 

7 

6 

9 

2 

5 

5 

8 

7 

6 

4 

5 

6 

9 

8 

5 

Bequired  the  sum 

20.  Of  6,  7,  5,  3,  2,  and  4. 

21.  Of  3,  2,  7,  4,  6,  and  7. 

22.  Of  3,  4,  5,  6,  7,  and  8. 

23.  Of  4,  5,  6,  2,  3,  and  5. 

24.  Of  2,  7,  3,  1,  4,  and  6. 

25.  Of  3,  5,  4,  3,  2,  and  4. 


26.  Of  3,  6,  7,  2,  1,  and  4. 

27.  Of  1,  3,  2,  7,  8,  and  5. 

28.  Of  8,  2,  4,  6,  5,  and  6. 

29.  Of  3,  6,  5,  3,  7,  and  2. 

30.  Of  4,  3,  2,  5,  6,  and  7. 

31.  Of  6,  2,  7,  4,  5,  and  8. 


2S.    CLvdSS   II— Problems   of  more   than   ona 
column. 

1.  What  is  the  sum  of  65,  46,  and  32? 


Solution  1 .  — We  write  the  numbers  units  under 
units  and  tens  under  tens,  and  commence  at  the 
right  to  add.  2  and  6  are  8  and  5  are  13,  units, 
which  equal  1  ten  and  3  units :  we  write  the  3 
units  under  the  column  of  units,  and  add  the  1 
ten  to  the  column  of  tens.  3  and  1  are  4  and  4 
are  8,  and  6  are  14,  tens,  which  equal  1  hundred 
and  4  tens ;  we  write  the  4  tens  in  tens'  place, 
and  the  1  hundred  in  hundreds'  place,  and  we 
have  143. 


OPERATION. 

65 
46 
32 


143  Ans. 


34  ADDITION. 

Solution  2. — After  the  pupil  is  familiar  with  the  above  solution 
he  may  abbreviate  it  thus:  2  and  6  are  8  and  5  are  13;  we  write 
the  3  and  add  the  1.  One  and  3  are  4,  and  4  are  8,  and  6  are  14, 
which  vTe  write. 


EXAMPLES   FOR   PRACTICE. 


(2.) 
43 

(3.) 

27 

(4.) 
37 

(5.) 
43 

(6.) 
58 

(7-) 
76 

38 

56 

25 

49 

36 

24 

81 

(8.) 
23 

(9.) 
28 

(10.) 
34 

(11.) 
44 

(12.) 

82 

(13.) 
18 

36 

51 

47 

56 

17 

71 

47 

35 

22 

31 

45 

49 

(14.) 
247 

(15.) 
462 

(16.) 
442 

(17.) 
756 

(18.) 
361 

(19.) 
826 

358 

379 

867 

482 

484 

108 

(20.) 
317 

(21.) 
424 

(22.) 
365 

(23.) 
813 

(24.) 

678 

<25.) 
725 

452 

536 

407 

791 

123 

146 

324 

817 

324 

142 

414 

234 

(26.) 
463 

(27.) 
282 

(28.) 
365 

(29.) 
216 

(30.) 
417 

(31.) 
318 

217 

187 

149 

418 

282 

182 

345 

208 

372 

732 

479 

479 

(32.) 
729 

(33.) 
321 

(34.) 
242 

(35.) 
813 

(36.) 
183 

(37.) 
815 

538 

467 

517 

916 

517 

581 

212 

213 

343 

732 

648 

186 

400 

457 

525 

145 

422 

307 

ADDITION. 

(38.) 

(39.) 

(40.) 

(41.) 

(42.) 

(43.) 

361 

217 

678 

678 

489 

289 

163 

721 

321 

910 

201 

303 

725 

548 

473 

112 

232 

132 

643 

918 

258 

814 

425 

333 

146 

172 

345 

756 

267 

456 

(44.) 

(45.) 

(46.) 

(47.) 

(48.) 

(49.) 

4567 

1718 

2526 

3343 

4243 

1525 

8910 

1920 

2728 

5363 

4546 

3545 

1112 

2122 

9303 

7389 

4748 

5565 

3456 

2324 

1323 

4041 

9505 

7585 

(50.) 

(51.) 

(52.) 

(53.) 

(54.) 

(55.) 

5960 

7374 

8163 

8124 

2185 

6215 

6162 

5789 

2738 

1792 

6727 

8372 

3646 

2100 

2543 

8547 

9858 

5728 

5666 

4731 

7342 

3218 

2832 

6217 

7869 

2578 

1856 

4002 

1479 

1234 

(56.) 

(57.) 

(58.) 

48721 

32 

173 

67321 

32578 

82573 

73214 

41625 

21 

289 

84366 

78321 

47020 

92785 

47856 

21832 

12346 

35 


59.  Find  the  sum  of  2185-f  6357+4832+6719+4324. 

60.  Find  the  sum  of  4344+4647+4849+5051+5253. 

61.  Find  the  sum  of  6432+7253+2187+6730+5087- 

62.  Find  the  sum  of  2426+3275+8397+2547+8037, 

63.  Find  the  sum  of  234+6721+853  +  8762+3739. 

64.  Find  the  sum  of  834+6737+8321  +  123+9207. 

65.  Find  the  sum  of  3246+2109+465+3712+2573 


36  ADDITION. 

66.  Find  the  sum  of  8213+123+6785+3282+7654. 

67.  Find  the  sum  of  123-f 456-f7821-f 6731+1234. 

68.  Find  the  sum  of  622+8763+1234+5678+910123. 

69.  Find  the  sum  of  23456+12345+70205+21846+ 
31082. 

PRACTICAL    PROBLEMS. 

1.  Mary  has  15  apples  and  John  has  23  apples:  how 
many  have  they  both  ? 

Solution. — If  Mary  lias  15  apples  and  John  operation. 

has  23  apples,  they  both  have  the  sum  of   15  15 

apples  and  23  apples,  which  is  38  apples.  23 

Note. — Very  young  pupils  may  say,  they  both  38  Ans. 

have  the  sum  of  15  apples  and  23  apples,  which  is 
38  apples. 

2.  There  were  25  robins  on  one  tree  and  36  robins  on 
another  tree;  how  many  robins  were  there  on  both 
trees  ? 

3.  AYiiiie  has  36  cents  in  one  pocket  and  45  cents  in 
the  other;  how  many  has  he  in  both  pockets? 

4.  A  little  boy  had  37  walnuts,  and  then  picked  56 
more  ;  how  many  walnuts  did  he  then  have  ? 

5.  Emma's  doll  cost  95  cents,  and  a  little  cradle  for  it 
cost  225  cents  j  how  much  did  both  cost  ? 

6.  There  were  48  roses  on  one  bush  and  39  roses  on 
another  bush ;  how  many  roses  were  there  on  both 
bushes? 

7.  A  little  girl  read  146  words  one  day  and  178  words 
the  next  day  j  how  many  words  did  she  read  both 
days  ? 

8.  Harry  had  246  cents  in  his  money-box,  and  his 
uncle  gave  him  175  cents;  how  many  cents  had  he 
then? 

9.  Peter's  kite  arose  436  feet,  and  Andrew's  kite  went 
58  feet  higher ;  how  high  did  Andrew's  kite  arise  ? 

10.  Edward  took  692  steps  in  going  to  school,  and 


ADDITION.  37 

Mary  took  742  steps ;  how  many  steps  did  they  both 
take  ? 

11.  Mary's  garden  contains  47  roses,  39  pinks  and 
52  lilies;  how  many  flowers  are  in  Mary's  garden? 

12.  Sallie  spelled  25  words  correctly,  Jennie  36  words, 
and  Maggie  28  words;  how  many  did  they  all  spell 
correctly  ? 

13.  Charlie  wrote  346  words  last  week  and  378  words 
this  week;  how  many  words  did  he  write  in  the  two 
weeks  ? 

14.  Minnie  saw  46  swallows  in  a  flock,  and  Maggie 
saw  54  swallows  in  another  flock;  how  many  swallows 
did  they  both  see  ? 

15.  Frank  says  he  took  627  steps  in  going  to  school, 
and  only  596  steps  in  coming  from  school ;  how  many 
steps  did  he  take  in  all  ? 

16.  My  father  has  6  horses,  13  cows,  and  46  sheep ; 
how  many  animals  has  he  in  all? 

17.  Emma's  new  reader  contains  46  pictures,  and 
Ella's  contains  78  pictures;  how  many  pictures  are 
there  in  both  of  these  readers  ? 

18.  Edward's  top  cost  25  cents,  his  whip  cost  43 
cents,  and  his  ball  cost  75  cents;  how  many  cents  did 
they  all  cost  ? 

19.  Albert's  father  owned  27  little  pigs,  and  Peter's 
father  owned  34  little  pigs ;  how  many  little  pigs  had 
they  both  ? 

20.  My  father  gave  215  cents  for  my  cap,  365  cents 
for  my  vest,  and  625  cents  for  my  coat ;  how  many  cents 
did  he  give  for  them  all  ? 

21.  One  old  hen  had  17  little  chickens,  another  had 
15  little  chickens,  and  another  16;  how  many  chickens 
did  the  three  hens  have  ? 

22.  Henry  learned  seventy-five  words  one  week  and 
eighty-four  words  the  next  week ;  how  many  words  did 
he  learn  both  weeks  ? 


88  ADDITION. 

23.  Maria  has  fifty-seven  cents  in  her  money-bank, 
and  her  aunt  put  twenty-five  cents  more  in  the  bank ; 
how  many  cents  did  she  then  have  ? 

24.  There  were  sixteen  robins  in  a  tree,  twenty-four 
on  the  barn,  and  thirty-nine  in  the  meadow;  how  many 
robins  were  there  in  all  ? 

25.  Julia  gave  a  poor  old  soldier  ninety-six  cents, 
Annie  gave  him  seventy-seven  cents,  and  Carrie  gave 
him  one  hundred  and  seventeen  cents ;  how  much  did 
the  old  soldier  receive  ? 

PRACTICAL   PROBLEMS. 

1.  A  gave  27  dollars  for  a  cow,  45  dollars  for  an  ox, 
and  150  dollars  for  a  horse ;  what  did  they  all  cost? 

2.  A  has  120  acres  of  land,  B  has  310  acres,  C  ha? 
516  acres,  and  D  has  715  acres;  how  many  acres  have 
they  together? 

3.  There  are  31  days  in  January,  28  in  February,  31 
in  March,  and  30  in  April ;  how  many  days  in  these 
four  months? 

4.  A  man  travelled  215  miles  one  week,  195  the  next 
week,  273  the  next,  and  378  the  next ;  how  far  did  he 
travel  ? 

5.  A  weighs  127  pounds,  B  weighs  215  pounds,  C 
176  pounds,  D  184  pounds,  and  E  234  pounds ;  what  is 
the  sum  of  their  weights  ? 

6.  A  farmer  raised  576  bushels  of  corn,  918  bushels 
of  oats,  3149  bushels  of  wheat,  and  2785  bushels  of  rye; 
how  many  bushels  did  he  raise  in  all? 

7.  A  owns  214  acres  of  land,  B  owns  719  acres,  C 
owns  2136  acres,  and  D  owns  372  acres ;  how  many 
acres  do  they  together  own  ? 

8.  A  bought  a  horse  for  168  dollars,  and  a  carriage 
for  376  dollars,  and  sold  them  so  as  to  gain  89  dollars; 
what  did  he  receive? 


ADDITION.  39 

9.  A  drover  had  327  sheep,  496  cows,  819  pigs,  123 
oxen,  and  216  horses  in  his  drove;  how  many  animals 
had  he  in  the  drove  ? 

10.  There  are  39  books  and  929  chapters  in  the  Old 
Testament,  and  37  books  and  260  chapters  in  the  ]S"ew 
Testament;  how  many  are  there  in  both? 

11.  In  an  orchard  87  trees  bear  apples,  26  bear 
peaches,  38  bear  plums,  and  17  bear  cherries;  how 
many  trees  are  there  in  the  orchard  ? 

12.  Mr.  Wilson's  farm  is  worth  3720  dollars,  his  bank 
stock  is  worth  1250  dollars,  and  he  has  7257  dollars  in 
money ;  how  much  is  he  worth  ? 

13.  A  man  bought  a  farm  for  7500  dollars,  paid  6550 
dollars  for  building  a  house  and  barn,  and  then  sold  it 
so  as  to  gain  725  dollars;  what  did  he  receive  for  it? 

14.  Harvey  bought  a  knife  for  37  cents,  a  hoop  for 
75  cents,  a  book  for  68  cents,  and  a  top  for  87  cents, 
he  sold  them  at  a  gain  of  23  cents ;  what  did  he  receive 
for  them  ? 

15.  William  lends  his  brother  3275  cents,  his  sister 
4287  cents,  his  father  3851  cents,  and  has  4892  cents 
left;  how  much  money  had  he? 

16.  In  one  book  there  are  725  pages,  in  another  book 
there  are  327  pages,  and  in  another  book  there  are  as 
many  as  in  both  the  former;  how  many  pages  in  all  ? 

17.  A  merchant  bought  cloth  for  756  dollars,  silk  for 
859  dollars,  muslin  for  367  dollars,  and  calico  for  255 
dollars ;  how  much  did  they  all  cost? 

18.  A  paid  325  dollars  for  a  span  of  horses,  and  248 
dollars  more  than  this  for  a  carriage;  for  how  much 
must  he  sell  them  both  to  gain  275  dollars? 

19.  A  gains  in  one  year  465  dollars,  B  gains  136 
dollars  more  than  A,  and  C  gains  as  much  as  A  and 
B  both;  how  much  did  B  gain?  how  much  did  G 
gain  ?  how  much  did  they  all  gain  ? 


40 


SUBTRACTION. 


SECTION   III. 
SUBTEACTIOK 

29.  Subtraction  is  the  process  of  finding  the  differ' 
ence  between  two  numbers. 

30.  The  Difference,  or  Remainder,  is  the  number 
of  units  more  in  the  greater  than  in  the  less. 

31.  The  Minuend  is  the  number  from  which  we  sub- 
tract.    The  Subtrahend  is  the  number  to  be  subtracted. 

32.  The  sign  of  Subtraction  is  — ,  and  is  read  minus. 
It  denotes  that  the  number  after  the  sign  is  to  be  sub- 
tracted from  the  one  before  it. 

Case  I. 

33.  To  subtract  when  no  figure  of  the  subtrahend 
expresses  more  units  than  its  figure  in  the  minu- 
end. 

34.  CLASS  I.— When  the  subtrahend  is  one 
figure. 

1.  Subtract  4  from  9. 


Solution. — We  write  the  4  under  the  9  and 
Bay,  4  units  from  9  units  leave  5  units,  which 
w;  write  beneath. 


OPERATION. 

9 

5  Ana. 


EXAMPLES    FOR   PRACTICE- 


(2) 

(3) 

(4.) 

(5.) 

(6.) 

(7.) 

5 

7 

5 

6 

8 

7 

2 

3 

3 

2 

5 

5 

(8) 

(9.) 

(10.) 

(11.) 

(12.) 

W 

8 

7 

6 

9 

8 

9 

3 

4 

3 

6 

2 

7 

SUBTRACTION. 


41 


14.  Subtract    3    from  9;    6    from    17;    7    from    19 ;    8 
from  19. 

15.  Subtract    2    from  14;    4   from  9;    8    from    18;    6 
from  19. 

16.  Subtract  7  from  18;    5  from  17;    6    from  18;    5 
from  16. 

17.  Subtract  8  from  19;    5  from  16;    9    from  19;    4 
from  17. 

35.  CLASS  II—  When  each  term  is  two  or  more 
figures. 

1.  Subtract  24  from  67. 


Solution — We  write  the  24  under  the  67,  units 
under  units,  and  tens  under  tens,  and  commence 
at  the  right  to  subtract.  4  units  from  7  units 
leave  3  units,  2  tens  from  6  tens  leave  4  tens; 
hence  the  remainder  is  4  tens  and  3  units,  or 
forty-three. 


OPERATION. 

67 
24 

43  Ans. 


EXAMPLES   FOR   PRACTICE. 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

58 

86 

72 

53 

46 

76 

35 

24 

41 

22 

15 

24 

(8.) 

(9.) 

(10.) 

(11.) 

(12.) 

(13.) 

49 

67 

85 

97 

86 

99 

27 

26 

52 

25 

73 

25 

(14.) 

(15.) 

(16.) 

(17.) 

(18.) 

(19.) 

625 

456 

763 

617 

767 

896 

312 

215 

512 

215 

123 

432 

(20.) 

(21.) 

(22.) 

(23.) 

(24.) 

(25.  J 

872 

725 

857 

907 

840 

876 

161 

413 

654 

205 

320 

345 

4* 


42 


SUBTRACTION. 

(26.) 

(27.) 

(28.)     (29.) 

(30.  j 

(31.) 

279 

807 

796    736 

967 

875 

136 

502 

452    432 

234 

345 

(32.) 

(33.) 

(34.)     (35.) 

(36.) 

(37.) 

8763 

9076 

3769    5076 

4872 

7659 

4321 

4054 

1546    3075 

2342 

3237 

(38.) 

(39.) 

(40.)     (41.) 

(42.) 

(43.) 

8769 

4876 

8275    8799 

8591 

6857 

3257 

2142 

3251    2542 

7230 

1234 

(44.  J 

(45.) 

(46.) 

(47.) 

(48) 

82345 

57596 

72578 

27397 

67385 

22121 

21321 

41362 

22315 

24123 

(49.) 

(50.) 

(51.) 

(52.) 

(53.) 

57897 

67858 

87578 

96754 

81296 

21472 

32721 

21335 

21423 

20135 

■ 
(54.) 

(55.) 

(56.) 

(57.) 

(58.) 

253786 

472589 

87695 

56728 

98785 

213123 

212423 

23542 

21306 

21342 

(59.) 

(60.) 

(61) 

(62.) 

(63.) 

373967 

873972 

72587 

95837 

89976 

212851 

132421 

51234 

51321 

32742 

Subtract 

64.  314  from  678. 

65.  425  from  658. 

66.  561  from  789. 

67.  254  from  576. 
18.  437  from  869. 


Subtract 

69.  1235  from  376$. 

70.  3726  from  4969. 

71.  2532  from  8748. 

72.  4720  from  87856. 

73.  12345  from  68799. 


SUBTRACTION.  43 

Case  II. 

36.  To  subtract  when  a  figure  in  tlie  subtrahend 
expresses  more  than  the  corresponding  figure  in 
the  minuend. 

37.  CLASS  I.— When  the  subtrahend  is  one 
figure. 

1.  Subtract  8  from  12.  operation. 

Solution. — We  write  the  8  under  the  12 ;  then  12 

8  from  twelve  is  four.  8 


4  Ans. 


EXAMPLES    FOR   PRACTICE. 


(20 
12 

(3.) 
12 

(4.) 
13 

(5.) 
14 

(6.) 
10 

11 

(8.) 

10 

(9.) 
13 

9 

7 

8 

6 

6 

7 

8 

9 
4 

(10.) 
15 

(11.) 
15 

(12.) 
16 

(13.) 
13 

(14.) 
16 

(15.) 
17 

(16.) 
16 

(17.) 
14 

7 

8 

9 

7 

8 

9 

6 

5 

(18.) 
10 

(19.) 
17 

(20.) 
13 

(21.) 
11 

(22.) 
10 

(23.) 
19 

(24.) 
14 

(25.) 
16 

3 

G 

5 

4 

2 

9 

8 

7 

3S.  CLASS  II.—  When  each  term  is  tivo  or  mors 
figures. 

1.  Subtract  45  from  82. 

Solution   1. — We    write    the   45   under    82,         operation. 
placing  units  under  units,  and  tens  under  tens,  82 

and  commence  at  the  right  to  subtract.     We  can-  45 

not  subtract  5  units  from  2  units;  we  will  there-  0_   . 

fore  take  1  ten  from  the  8  tens,  leaving  7  tens; 
1  ten  equals  10  units,  which  added  to  2  units 
equals  12  units;  5  units  from  12  units  leave  7  units;  4  tens  from  7 
tens  leave  3  tens;  hence,  the  remainder  is  37. 

Solution  2. — We  cannot  take  5  units  from  2  units;  we  will  there- 
fore add  10  units  to  the  2  units,  making  12  units;  5  units  from  13 


44  SUBTRACTION. 

units  leave  7  units.  Now,  since  we  have  added  10  units,  or  1  ten,  to 
the  minuend,  our  remainder  will  be  1  ten  too  large;  hence,  we  must 
add  1  ten  to  the  subtrahend;  1  ten  and  4  tens  are  5  tens,  5  tens  from 
8  tens  leave  8  tens. 

Note. — In  practice  we  solve  thus;  5  from  2  we  cannot  take,  but  5  from 
12  leaves  7,  4  and  1  are  5,  and  5  from  8  leaves  3. 

39.  From  the  preceding  explanations  we  have  the 
following  general  rule. 

Eule. — 1.  Write  the  smaller  number  under  the  larger, 
with  units  under  units,  tens  under  tens,  etc.,  and  commence 
at  the  right  to  subtract. 

2.  Take  the  number  denoted  by  each  figure  of  the  subtra- 
hbnd  from  the  number  denoted  by  the  corresponding  figure 
of  the  minuend,  and  write  the  result  beneath. 

3.  If  the  number  denoted  by  a  figure  in  the  subtrahend  is 
greater  than  the  number  denoted  by  the  corresponding  figure 
in  the  minuend,  add  10  to  the  latter  and  then  subtract,  and 
add  1  to  the  next  left-hand  place  in  the  subtrahend. 

40.  Proof. — Add  cne  remainder  to  the  subtrahend ; 
the  sum  will  equal  the  minuend  if  the  work  is  correct. 

EXAMPLES   FOR   PRACTICE. 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

73 

64 

32 

41 

53 

62 

25 

27 

14 

26 

28 

28 

(8.) 

(9.) 

(10.) 

(11.) 

(12.) 

(13.) 

75 

31 

57 

63 

87 

95 

26 

18 

29 

45 

28 

59 

(14.) 

(15.) 

(16.) 

(17.) 

(18.) 

(19.) 

87 

75 

63 

77 

87 

94 

39 

38 

25 

48 

59 

49 

(20.) 

(21.) 

(22.) 

(23.) 

(24.) 

(25.) 

72 

84 

70 

81 

90 

97 

27 

48 

17 

18 

39 

79 

(26.) 
342 

(27.) 
573 

(28.) 
692 

(29.) 
545 

(30.) 
826 

(81.) 
357 

124 

245 

457 

328 

252 

183 

(32.) 
573 

(33.) 

748 

(34.) 
835 

(35.) 
968 

(36.) 
839 

(37.) 
538 

248 

375 

573 

675 

584 

394 

(38.) 
659 

(39.) 
839 

(40.) 
547 

(41.) 
658 

(42.) 

735 

(43.) 
848 

475 

583 

284 

372 

373 

539 

(44.) 
524 

(45.) 
752 

(46.) 
845 

(47.) 
307 

(48.) 
456 

(49.) 
450 

356 

387 

579 

138 

387 

382 

(50.) 
854 

(61.) 

943 

(52.) 
607 

(53.) 
500 

(54.) 

704 

(55.) 
403 

396 

765 

309 

325 

507 

285 

(56.) 
726 

(57.) 

857 

(58.) 

735 

(59.) 
792 

(60.) 

807 

(61.) 
650 

387 

389 

558 

295 

328 

357 

(62.) 

3876 

(63.) 
6385 

(64.) 
6735 

(65.) 

4076 

(66.) 
4070 

(67.) 
4135 

2379 

3527 

2547 

3128 

2137 

1216 

(68.) 
8672 

(89.) 

5283 

(70.) 
8175 

(71.) 
2534 

(72.) 
6735 

(73.) 
7219 

3728 

2426 

2836 

1235 

5376 

1972 

(74.) 
8522 

(75.) 
7135 

(76.) 
6347 

(77.) 
8135 

(78.) 
7345 

(79.) 
4372 

6243 

1872 

2503 

£453 

2876 

2583 

(80.) 
35672 

(81.) 
43763 

(82.) 
87253 

(83.) 
73875 

(84.) 
63527 

(86.) 

53413 

23828 

24235 

34365 

38376 

14238 

28401 

45 


46 


SUBTRACTION. 

(86.) 

73285 

(87.) 
20307 

(88.) 
87004 

(89.) 
76500 

(90.) 
20500 

(91.) 
37201 

43836 

15231 

34523 

43654 

37254 

23534 

(92.) 
83030 

(93.) 
90304 

(94.) 
50310 

(95.) 
60204 

(96.) 
70000 

(97.) 
100000 

76513 

40372 

30311 

30205 

32463 

1 

PRACTICAL    PROBLEMS. 
1.  Mary  had  25  roses  and  gave  Sarah  12  of  them; 
how  many  did  Mary  then  have  ? 


OPERATION. 

25 
12 

13  Ans. 


Solution. — If  Mary  had  25  roses  and  gave 
Sarah  12  of  them,  Mary  then  had  the  difference 
between  25  roses  and  12  roses,  which  is  13  roses. 

Note. — Quite  young  pupils  may  say,  Mar}7  then 
had  the  difference  between  25  roses  and  12  roses, 
which  is  13  roses. 

2.  Willie  had  34  cents  and  gave  James  18  cents;  how 
many  cents  did  Willie  then  have? 

3.  A  little  girl  had  54  pins  and  gave  her  cousin  27 
of  them-  how  many  did  she  have  remaining? 

4.  Fifty  little  robins  were  sitting  on  a  tree,  and  23  of 
them  flew  away  j  how  many  were  then  left  ? 

5.  There  were  96  peaches  on  an  old  peach-tree,  and  a 
gust  of  wind  blew  37  of  them  off;  how  many  then  re- 
mained on  the  tree  ? 

6.  Henry's  top  and  ball  cost  120  cents;  how  much 
did  the  top  cost,  if  the  ball  cost  75  cents  ? 

7.  Emma's  doll  and  its  little  cradle  cost  320  cents, 
and  the  doll  cost  95  cents ;  how  much  did  the  cradle  cost  ? 

8.  There  were  87  roses  on  two  rose-bushes;  how 
many  roses  were  there  on  the  second  bush,  if  there  were 
39  roses  on  the  first  bush  ? 

9.  A  little  girl  read  324  words  in  two  days ;  if  she 
read  146  words  one  day,  how  many  did  she  read  the 
other  day  ? 


SUBTRACTION.  47 

10.  Edward  and  Mary  together  took  1434  steps  in 
going  to  school;  how  many  steps  did  Mary  take,  if 
Edward  took  692  steps  ? 

11.  Minnie  had  372  cents  in  her  money-bank,  and  took 
out  164  cents  to  give  to  a  little  beggar-girl  j  how  many 
cents  remained  ? 

12.  Andrew's  kite  arose  494  feet,  and  this  was  58  feet 
higher  than  Peter's  kite  went ;  how  high  did  Peter's 
kite  fly  ? 

13.  Charlie  wrote  724  words  in  two  weeks;  he  wrote 
346  words  the  first  week;  how  many  words  did  ho 
write  the  second  week  ? 

14.  Mary's  new  reader  contains  76  pictures,  and 
Fanny's  contains  92  pictures;  how  many  does  Fanny's 
contain  more  than  Mary's  ? 

15.  Two  little  girls  picked  74  quarts  of  blackberries 
one  summer;  if  one  picked  37  quarts,  how  many  quarts 
did  the  other  pick  ? 

16.  Thomas  said  he  counted  283  crows  in  his  father's 
cornfield ;  he  threw  a  stone  and  scared  126  away ;  how 
many  then  remained  ? 

17.  Floy  and  Eugie  together  took  3000  steps  one 
day  ;  if  Floy  took  1786  steps,  how  many  steps  did  Eugie 
take  ? 

18.  Effie  and  Eddie  counted  their  chestnuts  and  found 
they  together  had  1232 ;  now,  if  Eddie  had  675,  how 
many  had  Effie  ? 

19.  Mary's  mother  bought  her  an  arithmetic  and 
slate  for  125  cents ;  if  the  slate  cost  45  cents,  what  did 
the  arithmetic  cost  ? 

20.  Herbert's  father  bought  him  a  cap  and  coat  for 
850  cents;  he  paid  125  cents  for  the  cap;  how  much 
did  he  pay  for  the  coat  ? 

21.  Mr.  Xelson's  horse  and  carnage  cost  four  hundred 
dollars ;  what  did  the  horse  cost,  if  the  carriage  cost 
two  hundred  and  twenty-five  dollars  ? 


iS  SUBTRACTION. 

22.  Two  little  girls  picked  seventy-four  quarts  of 
blackberries  one  summer;  if  one  picked  thirty-seven 
quarts,  how  many  quarts  did  the  other  pick? 

23.  Mr.  Barton  raised  two  thousand  bushels  of  wheat 
and  rye ;  how  much  rye  did  he  raise,  if  he  raised  five 
hundred  and  sixty-five  bushels  of  wheat  ? 

PRACTICAL   PROBLEMS. 

1.  A  man  had  78  cows  and  sold  24  of  them;  how 
many  cows  remained  ? 

Solution. — If  a  man  had  78  cows  and  sold         operation. 
24,   there  remained  the  difference  between  78  78 

and  24,  which  we  find  by  subtracting  is  54.  24 

Note. — Quite  young  pupils  may  merely  say,  there  54  Ans. 

remains  the  difference  between  78  and  24,which  is  54. 

2.  A  boy  had  150  cents  and  spent  75  cents;  how 
many  cents  then  remained  ? 

3.  A  man  had  325  apples  and  sold  180  apples ;  how 
many  had  he  then  ? 

4.  Henry  had  1735  dollars,  lent  his  brother  854  dol- 
lars ;  how  man}''  dollars  remained  ? 

•5.  A  bought  570  horses  and  sold  295  of  them;  how 
many  remained  unsold  ? 

6.  A  and  B  together  had  7256  acres  of  land;  how 
many  had  B  if  A  had  3627  ? 

7.  Two  men  have  8570  bushels  of  grain,  and  the  first 
has  2846  bushels ;  how  many  has  the  second  ? 

8.  Washington  was  born  1732  and  died  1799;  how 
old  was  he  at  his  death  ? 

9.  John  Adams  was  born  1735  and  died  1826;  how 
old  was  he  at  his  death  ? 

10  Jefferson  was  born  1743  and  died  1826;  how  old 
was  he  at  his  death  ? 

11.  Madison  was  born  1758  and  died  1836;  how  old 
was  he  at  his  death  1 


SUBTRACTION.  49 

12.  Monroe  was  born  1758  and  died  1831  j  how  old 
was  he  at  his  death  ? 

13.  John  Quincy  Adams  was  born  1767  and  died  1848; 
now  old  was  he  at  his  death  ? 

14.  Jackson  was  born  1767  and  died  1845;  how  old 
was  he  at  his  death  ? 

15.  A  has  5480  bushels  of  oats,  which  is  975  bushels 
more  than  B  has ;  how  many  bushels  has  B  ? 

16.  In  an  army  of  50000  men  628  were  killed  and 
2596  wounded;  how  many  remained  unhurt? 

17.  A  farmer  had  234  hens  and  bought  367,  and  then 
sold  489 ;  how  many  then  remained. 

18.  A  merchant  sold  goods  to  the  amount  of  7580 
dollars  and  gained  1396  dollars;  what  did  the  goods 
cost? 

19.  A  and  B  have  each  1840  acres  of  land ;  A  sells  B 
895  acres ;  how  many  has  each  then  ? 

20.  A  farmer  has  1346  sheep  and  849  lambs;  how 
many  more  sheep  has  he  than  lambs  ? 

21.  Mary  and  Eliza  have  each  789  cents;  if  Eliza 
gives  Mary  247  cents,  how  many  will  each  then  have  ? 

22.  Subtract  six  hundred  and  seventy-eight  from  nine 
hundred  and  four. 

23.  Add  seven  hundred  and  fifteen  to  five  hundred 
and  seventy-three,  and  subtract  the  sum  from  two 
thousand. 

24.  Find  the  sum  of  one  thousand  and  ninety-six  and 
five  hundred  and  forty-five,  and  subtract  it  from  three 
thousand. 

25.  Frank  solved  four  hundred  and  sixteen  problems, 
and  Fanny  solved  five  hundred  and  three  problems; 
how  many  did  Fanny  solve  more  than  Frank  ? 

26.  A  farmer  had  2346  bushels  of  wheat ;  he  sold  one 
man  687  bushels  and  another  man  1560  bushels;  how 
many  bushels  did  he  sell  ?  how  many  remained  ? 

5 


50  PRACTICAL    PROBLEMS. 

PRACTICAL    PROBLEMS 
in  Addition  and  Subtraction. 

1.  If  I  have  75  cents  in  my  money -bank,  and  my 
uncle  puts  in  26  cents,  how  much  will  be  in  it  then  ? 

2.  If  Willie  reads  125  words  this  week  and  187  words 
next  week,  how  many  words  will  he  read  in  all  ? 

3.  My  father  had  236  little  chickens,  and  a  mink  killed 
48  of  them  ;  how  many  remained  ? 

4.  If  I  have  438  dollars  and  give  my  sister  246  dol- 
lars, how  much  will  I  have  remaining? 

5.  Mary's  father  had  360  acres  of  land  and  sold  125 
acres  ;  how  many  acres  did  he  then  have  ? 

6.  Peter  had  467  dollars  and  lent  his  brother  185  dol- 
lars ;  how  much  did  he  then  have  ? 

7.  I  have  365  cents  in  my  money-bank ;  how  many 
must  I  put  in  that  there  may  be  400  cents  in  it  ? 

8.  Sallie  had  72  cents  and  her  brother  gave  her 
enough  to  make  her  money  134  cents ;  how  much  did  her 
brother  give  her  ? 

9.  Carrie's  brother  teased  her  because  she  couldn't 
tell  how  many  she  must  add  to  245  to  make  400 ;  can 
you  tell  ? 

10.  Matilda  had  120  cents,  her  mother  gave  her  236 
cents,  and  then  she  lent  her  brother  248  cents  ;  how 
many  cents  did  she  then  have  ? 

11.  Fannie  picked  236  chestnuts,  her  little  brother 
gave  her  78  chestnuts,  and  she  gave  95  to  her  school- 
mates ;  how  many  chestnuts  remained  ? 

12.  Mary  cried  because  she  couldn't  tell  her  teacher 
how  many  she  must  add  to  367  to  make  500 ;  tell  me 
how  many  it  is. 

13.  One  morning  in  going  to  school  I  took  726  steps ; 
how  many  more  would  I  have  taken  if  I  had  taken 
1000  in  all  ? 

14.  My  kite  was  up  in  the  air  436  feet,  it  then  fell 
185  feet,  and  then  arose  260  feet ;  how  high  was  it  then  ? 


BUSINESS    PROBLEMS.  51 

BUSINESS   PROBLEMS. 

1.  I  went  to  a  store  mid  bought  a  book  for  87  cents 
and  a  slate  for  35  cents ;  what  did  I  pay  for  both  of 
them  ? 

2.  My  mother  took  me  to  a  store  and  bought  me  a  top 
for  15  cents,  a  cap  for  75  cents,  and  a  knife  for  45  cents ; 
what  did  they  all  cost  ? 

3.  William's  slate  cost  26  cents,  his  arithmetic  55  cents, 
his  reading-book  48  cents,  and  his  spelling-book  37 
cents  ;  what  did  they  all  cost  ? 

4.  I  went  to  a  store  and  bought  a  knife  for  56  cents 
and  gave  the  storekeeper  a  dollar  bill  (100  cents)  to  pay 
for  it ;  how  much  change  did  he  give  me  back  ? 

5.  Mary  bought  a  flower-vase  for  375  cents,  and  handed 
the  storekeeper  a  five-dollar  bill  (500  cents)  to  pay  for 
it ;  how  much  change  should  she  have  received  ? 

6.  Mr.  Barnes  paid  75  dollars  for  his  watch,  and  sold 
it  so  that  he  gained  12  dollars ;  what  did  he  receive 
for  it  ? 

7.  Martha's  new  shawl  cost  875  cents ;  if  she  should 
sell  it  so  as  to  gain  125  cents,  what  would  she  receive 
for  it  ? 

8.  Mr.  Taylor's  new  house  cost  him  3675  dollars,  and 
he  sold  it  for  565  dollars  more  than  it  cost  him ;  what 
did  he  receive  for  it  ? 

9.  My  father  bought  a  cow  for  38  dollars  and  sold 
her  for  52  dollars;  how  much  did  he  gain  on  the  cow  ? 

10.  Robert  Stewart  had  a  coat  which  cost  him  45  dol- 
lars ;  he  sold  it  to  Edward  Taylor  for  37  dollars ;  how 
much  did  he  lose  ? 

11.  Harry  Hartman  sold  his  watch  for  67  dollars  and 
lost  by  the  sale  15  dollars  ;  what  did  the  watch  cost  him  ? 

12.  Mary's  papa  gave  her  a  5  dollar  bill  to  go  a  shop- 
ping; she  bought  a  fan  for  75  cents,  somo  silk  for  165 
cents,  and  a  pair  of  gloves  for  125  cents;  how  much 
change  did  she  bring  home  ? 


32  SUBTRACTION, 

PRACTICAL   PROBLEMS 
in  Addition  and  Subtraction. 

1.  What  is  the  value  of  675  +  432  +  285  +  672? 

2.  What  is  the  value  of  362  +  486  +  721  —  367  ? 

3.  What  is  the  value  of  473  +  325  +  604—1206? 

4.  What  is  the  value  of  3072  +  4861  +  2075  —  6785? 

5.  Subtract  1678  from  the  sum.  of  985  and  863. 

6.  Subtract  the  sum  of  265  and  381  from  the  sum  of 
281  and  678. 

7.  Subtract  218  +  318  +  418  from  379  +  279+479. 

8.  A  having  475  dollars  earned  220  dollars  and  then 
spent  567  ;  how  much  remained  ? 

9.  Newspapers  were  first  published  in  1630;  how 
long  have  they  been  published  ? 

10.  Quills  were  first  used  for  writing  about  the  year 
636;  how  long  is  it  since? 

11.  Cotton  was  first  planted  in  the  United  States 
about  the  year  1769  j  how  many  years  since  ? 

12.  Glass  windows,  it  is  said,  were  first  used  in  Eng- 
land in  1180  ;  how  long  is  it  since  then  ? 

13.  A  sold  his  farm  for  12450  dollars,  which  was  1680 
dollars  more  than  it  cost;  howjnuch  did  it  cost? 

14.  A  gave  6500  dollars  for  his  farm  and  2560  dollars 
for  his  house,  and  sold  them  for  12000;  what  was  the 
gain  ? 

15.  A  farmer  had  5600  bushels  of  corn,  and  sol'd  1850 
bushels  to  A  and  2810  to  B ;  how  much  remained? 

16.  The  area  of  Maine  is  30000  square  miles,  and  of 
New  York  46000 ;  how  much  larger  is  New  York  than 
Maine  ? 

17.  The  area  of  Massachusetts  is  7800  square  miles, 
and  of  Pennsylvania  46000  square  miles ;  how  much 
larger  is  Pennsylvania  than  Massachusetts  ? 

18.  How  much  larger  are  Pennsylvania  and  Maine 
together  than  New  York  and  Massachusetts  together? 


MULTIPLICATION.  53 


MULTIPLICATION. 

41.  Multiplication  is  the  process  of  finding  the  re- 
sult of  taking  one  number  as  many  times  as  there  aro 
units  in  another.    . 

42.  The  Multiplicand  is  the  number  to  be  multiplied. 

43.  The  Multiplier  is  the  number  by  which  we 
multiply. 

44.  The  Product  is  the  result  obtained. 

44£.  The  sign  of  Multiplication  is  X  >  and  is  read 
multiplied  by:  thus,  4  X  3  =  12  means  4  multiplied  by  3 
equals  12.  The  4  is  the  multiplicand,  3  is  the  multiplier, 
and  12  is  the  product. 

Note  to  Teachers. — If  the  pupils  are  not  familiar  with  the  Mul- 
tiplication Table,  let  them  now  turn  to  page  20  and  learn  it. 

Case  I. 

45.  When  the  multiplier  is  one  figure. 

CLASS  I. —  When  no  product  exceeds  nine, 
1.  Multiply  34  by  2. 

Solution. — We  write  the  multiplier  under  the  operation. 
multiplicand,  and  begin  at  the  right  to  multiply.  34 

2  times  4  ufcits  are  8  units;  we  write  the  8  units  2 

in  units'  place.     2  times  3  tens  are  6  tens  ;  we  qq  Ans. 

write  the  6  tens  in  tens'  place. 

(2.)  (3.)  (4.)  (5.) 

32  24  14  41 

2  2  2  2 

(6.)  (7.)  (8.)  (9.) 

21  12  23  32 

3  3  3  3 

40.  CLvdSS  II—  When  some  of  the  products  ex- 
ceed nine. 

1.  Multiply  56  by  4. 

5* 


51  MULTIPLICATION. 

Solution  1. — We  write  the  multiplier  under         operation. 
the  multiplicand  and  begin  at  the  right  to  multi-  56 

ply.    4  times  6  units  are  24  units,  which  equal    4  _4 

units  and  2  tens.     We  write  the  4  units  in  units  £24  Ans. 

place,  and  reserve  the  2  tens  to  add  to  the  next 
product.     4  times  5  tens  are  20  tens,  plus  the  2  tens  equal  22  tens, 
which  equal  2   tens  and  2  hundreds,  which  we  write  in  their  proper 
places.     Hence  the  product  is  224. 

Solution  2. — 4  times  6  are  24;  we  write  the  4  and  add  the  2  to 
the  next  product.  4  times  5  are  20,  and  2  added  equal  22 ;  hence 
the  product  is  224.     From  this  we  have  the  following 

Eule. —  Write  the  multiplier  under  the  multiplicand, 
draw  a  line  beneath,  begin  at  units,  and  multiply  the  num- 
her  denoted  by  each  figure  of  the  multiplicand  by  the  multi- 
plier, carrying  as  in  addition. 


(2-) 
25 

(3.) 
36 

(4.) 
47 

(5.) 
73 

(6.) 

28 

3 

2 

3 

2 

4 

63 

(8.) 

75 

(9.) 

36 

(10.) 
27 

(11.) 
43 

5 

4 

5 

6 

5 

— 

— 

— 

— — 

■ 

(12.) 

75 
2 

(13.) 

86 
3 

(14.) 
92 
4 

(15.) 

76 
5 

(16.) 
,84 
6 

(17.) 
73 

(18.) 
47 

(19.) 
76 

(20.) 

85 

(21.) 

73 

5 

6 

7 

8 

8 

— 

— 

— 

— 

■ 

(22.) 
234 

(23.) 
425 

(24.) 
673 

(25.) 
723 

(26.) 
351 

3 

4 

5 

6 

7 

(27.) 
425 

(28.) 
314 

(29.) 
421 

(30.) 
636 

(31.) 

854 

6 

7 

8 

7 

3 

MULTIPLICATION. 


(32.) 

(33.) 

(34.) 

(35.) 

(36.) 

256 

375 

873 

358 

725 

4 

6 

7 

8 

7 

(37.) 

(38.) 

(39.) 

(40.) 

(41.) 

581 

809 

394 

908 

765 

7 

8 

6 

9 

8 

Multiply 


42 

3124  by  4. 

43. 

2856  by  5. 

44. 

7863  by  6. 

45. 

2185  by  7. 

46. 

4182  by  8. 

47. 

3075  by  8. 

48. 

4107  by  9. 

49. 

7685  by  6. 

Multiply 

50.  13257  by  2. 

51.  36072  by  3. 

52.  85761  by  4. 

53.  35167  by  5. 

54.  84307  by  6. 

55.  30754  by  7. 

56.  21836  by  8. 

57.  35168  by  9. 


Case  II. 
47.  Wlien  the  multiplier  consists  of  two  or  more 
figures. 

4S.  CLASS  L—  When  the  multiplier  consists  of 
two  jig  ures. 

1.  Multiply  64  by  23. 


Solution  1. — We  write  the  multiplier  under 
the  multiplicand,  placing  units  under  units,  and 
tens  under  tens,  and  begin  at  the  right  to  mul- 
tiply. 3  times  4  units  are  12  units,  which  equals 
1  ten  and  2  units ;  we  write  the  units  under  the 
3,  and  reserve  the  1  ten  to  add  to  the  next  pro- 
duct. 3  times  6  tens  are  18  tens,  and  1  ten 
added  equals  19  tens,  or  1  hundred  and  9  tens,  which  we  write  in 
their  proper  places.  Multiplying  64  by  2  in  the  same  manner,  we 
have  128,  and  since  the  2  is  2  tens  we  have  128  tens,  which  we  write 
m  its  proper  place  ;  then,  adding  the  two  products,  we  have  1472. 

Solution  2.— Three  times  4  are  12;  we  write  the  2  and  carry  the 


OPERATION. 

64 
23 

192 
128 

147li  Ans. 


56  MULTIPLICATION. 

1  :  three  times  6  are  18,  plus  the  1  equals  19 ;  which  we  write. 
Then,  2  times  4  are  8,  which  we  write  uuder  the  2,  and  2  times  6 
are  12,  which  we  write  beside  the  8. 

Bule. — 1.  Write  the  multiplier  under  the  multiplicand, 
placing  units  under  units,  tens  under  tens  etc.,  and  begin  at 
the  right  to  multiply. 

2.  Multiply  the  multiplicand  by  the  number  denoted  by 
each  figure  of  the  multiplier-,  writing  the  first  figure  of  each 
product  under  the  figure  of  the  multiplier  used. 

3.  Add  togethej  the  partial  products,  and  their  sum  will 
be  the  entire  product. 

Proof. — Multiply  the  multiplier  by  the  multiplicand  ; 
if  the  two  results  ag*'ee,  the  work  is  probably  correct 


(2.) 
38 

(3.) 
43 

73 

(5.) 
81 

(6.) 
29 

(7-.) 

57 

23 

24 

35 

67 

82 

75 

(8.) 
87 

(9.) 
39 

(10.) 

87 

(11.) 
29 

(12.) 
123 

(13.) 
245 

28 

43 

52 

92 

37 

32 

(14.) 
436 

(15.) 
534 

(16.) 
427 

(17.) 
426 

(18.) 
534 

(19.) 
672 

43 

43 

35 

43 

45 

46 

(20.) 
725 

(21.) 
634 

(22.) 
807 

(23.) 
475 

(24.) 

709 

(25.) 
493 

42 

47 

37 

54 

88 

82 

(26.) 
756 

(27.) 
762 

(28.) 
675 

(29.) 
467 

(30.) 

762 

(31.) 

812 

93 

48 

39 

37 

62 

45 

MULTIPLICATION. 


57 


(32.) 

(33.)     (34.) 

(35.) 

(36.)     (37.) 

1234 

2341    6724 

6357 

7138    2536 

28 

35     42 

35 

52     25 

(38.) 

(39.)     (40.) 

(41.) 

(42.)     (43.) 

6347 

8192    4736 

4825 

3121    4073 

46 

73     63 

72 

37     46 

Multiply 

Multiply 

44. 

6538  by  83. 

49. 

4175  by  28. 

45. 

7384  by  45. 

50. 

7186  by  85. 

46. 

2185  by  67. 

51. 

8391  by  94. 

47. 

3407  by  82. 

52. 

2187  by  89. 

48. 

3584  by  46. 

53. 

6543  by  98. 

49.   CLASS  II.— When  the  multiplier  consists 
of  three  figures. 


(1.) 

4126 

(2-) 
5731 

(3.) 
7351 

(4) 
1375 

(5.) 
5379 

234 

243 

432 

342 

423 

(6.) 
6725 

(7.) 
2183 

(8.) 
7321 

(9.) 
8193 

(10.) 
2147 

345 

544 

265 

4/o 

813 

(11.) 
2143 

(12.) 
8192 

(13.) 
2435 

(14.) 
4167 

(15.) 
8246 

227 

426 

146 

245 

642 

(16.) 
7346 

(17.) 
7516 

(18.) 
8927 

(19.) 
4928 

(20.) 
2076 

643 

571 

352 

816 

437 

r>8 


MULTIPLICATION. 

(21.)       (22.) 

(23.) 

(24.)      (25.) 

4752     7385 

8492 

2937     6473 

185      218 

537 

439      567 

Multiply 

Multiply 

26.  14651  by  283. 

35. 

28352  by  345. 

27,  31251  by  625. 

36. 

41678  by  287. 

28.  36782  by  234. 

37. 

34073  by  435. 

29.  43678  by  452. 

38. 

40735  by  628. 

30.  36507  by  634. 

39. 

29304  by  789. 

31.  40725  by  365. 

40. 

90705  by  897. 

32.  32107  by  681. 

41. 

43445  by  678. 

33.  25697  by  329. 

42. 

37436  by  835. 

34.  42046  by  456. 

43. 

88888  by  789. 

50.  CLASS  III.— When  the  multiplier  consists 
of  more  than  three  figures. 


4137 

(2.) 
3642 

(3.) 
6724 

(4-) 
4183 

(5.) 
3645 

(6.) 
4526 

2185 

2531 

3625 

2426 

2841 

2182 

3482 

(8.) 
2846 

(9.) 

3707 

(10.) 
4172 

(11.) 

2882 

(12.) 
8567 

2534 

2528 

2851 

2174 

2773 

3178 

(13.) 

5185 

(14.) 
9187 

(15.) 

4785 

(16.) 

8197 

(17.) 
4376 

(18.) 
8765 

8763 

2567 

7372 

1846 

5273 

5678 

Multiply 

19.  28751  by  3146. 

20.  17346  by  2435. 

21.  21307  by  3147. 

22.  85276  by  3452. 


Multiply 

23.  72509  by  3167. 

24.  85216  by  2431. 

25.  73519  by  4735. 

26.  81897  by  3456. 


MULTIPLICATION. 


59 


Multiply 

Multiply 

27. 

21346  by  31452. 

31. 

10786  by  31672. 

28. 

47309  by  45233. 

32. 

47396  by  73462. 

29. 

25737  by  63252. 

33. 

76448  by  54173. 

30. 

43629  by  28516. 

34. 

28354  by  31867. 

51.   CLASS  IV.— When  one  or  both  terms  con- 
tain ciphers. 

1.  Multiply  5721  by  3006-  also,  37000  by  2400. 


OPERATION. 

5721 

3006 


OPERATION. 
37000 

2400 


34326 
17163 

17197326 


148 
74 


88800000 


Note. — In  the  first  example,  pass  over  the  naughts,  placing  the  right- 
hand  figure  of  the  product  by  3  directly  under  the  3.  In  the  second 
problem,  we  multiply  by  the  significant  figures,  and  then  annex  the 
naughts  to  the  product. 


Multiply 

Multiply 

2. 

3678  by  204. 

11. 

4500  by  2800. 

3. 

4107  by  307. 

12. 

67000  by  450. 

4. 

4178  by  1005. 

13. 

96000  by  2800. 

5. 

8675  by  3007. 

14. 

87000  by  4800. 

6. 

7276  by  6008. 

15. 

73500  by  32000. 

7. 

4136  by  2305. 

16. 

86700  by  47200. 

8. 

8449  by  3046. 

17. 

32800  by  346000. 

9. 

4592  by  5607. 

18. 

70900  by  407100. 

to 

8124  by  4801. 

19. 

85900  by  1030600 

rO.  Multiply  four  thousand  six  hundred  and  ten  by 
seven  thousand  and  fort}-. 


60  .MULTIPLICATION. 

EXAMPLES   IN   MULTIPLICATION. 

1.  If  one  orange  cost  8  cents,  what  will  7  oranges  cost 
at  the  same  rate  ? 

OPERATION. 

Solution. — If    one   orange   cost   8   cents,    7  8 

oranges  will  cost  7  times  8  cents,  which  are  56  7 

cents.  56  Ans. 

2.  If  one  pig  cost  7  dollars,  what  will  6  pigs  cost  at 
the  same  rate  ? 

3.  If  a  yard  of  muslin  cost  37  cents,  what  will  8  yards 
cost  at  the  same  rate  ? 

4.  If  a  boy  writes  36  words  in  a  day,  how  many  will 
he  write  in  13  days  ? 

5.  What  must  I  pay  for  15  cows,  if  I  pay  28  dollars 
for  each  cow  ? 

6.  If  Henry  takes  42  steps  in  a  minute,  how  many 
steps  will  he  take  in  15  minutes  ? 

7.  If  a  car  runs  25  miles  in  an  hour,  how  far  will  it 
run  in  12  hours  ? 

.  8.  If  a  boy  learns  14  new  words  each  clay,  how  many 
will  he  learn  in  11  days  ? 

9.  Mary  has  14  rose-bushes  in   her   garden,  and  on 
each  bush  there  are  26  roses ;  how  many  roses  on  all  ? 

10.  How  much  must  I  pay  for  16  pounds  of  tea,  at 
the  rate  of  78  cents  a  pound? 

11.  What  are  nine  loads  of  hay  worth,  at  the  rate  of 
23  dollars  a  load  ? 

12.  If  one  cord  of  wood  is  worth  six  dollars,  how 
much  are  18  cords  of  wood  worth  ? 

13.  How  many  marbles  will  7  boys  have,  if  each  boy 
has  12  marbles  ? 

14.  At  the  rate  of  45  miles  a  day,  how  far  will  a  per- 
son travel  in  23  days  ? 

15    If  Henry  can  count  65  m  a  minute,  how  taany 
can  he  count  in  26  minutes  ? 


MULTIPLICATION.  61 

PRACTICAL  EXAMPLES 
in  Multiplication. 

1.  What  cost  24  horses  at  245  dollars  each  ? 

Solution. — If  one  "horse  cost  245  dollars,  24  horses  cost  24  times 
245  dollars,  which  by  multiplying  we  find  to  be  5880  dollars. 

2.  If  a  boat  sails  246  miles  in  one  day,  how  far  will  it 
sail  in  26  days? 

3.  If  in  one  book  there  are  364  pages,  how  many 
pages  in  18  such  books  ? 

4.  In  one  barrel  of  flour  there  are  196  pounds ;  how 
many  pounds  in  25  barrels  of  flour  ? 

5.  How  much  will  42  horses  cost,  at  the  rate  of  150 
dollars  apiece  ? 

6.  If  an  acre  of  land  is  worth  218  dollars,  how  much 
will  76  acres  cost  ? 

7.  If  in  an  orchard  there  are  32  rows  of  trees  with 
46  trees  in  a  row,  how  many  trees  in  all  ? 

8.  A  man  bought  326  horses  and  36  times  as  many 
sheep  ;  how  many  sheep  did  he  buy  ? 

9.  What  cost  125  yards  of  cloth  at  the  rate  of  325 
cents  a  yard  ? 

10.  How  much  will  236  bushels  of  wheat  cost,  at  175 
cents  a  bushel? 

11.  There  are  1760  yards  in  one  mile;   how  many 
yards  in  12  miles  ? 

12.  There  are  5280  feet  in  a  mile;  how  many  feet  in 
18  miles? 

13.  There  are  660  feet  in  one  furlong ;  how  many  feet 
in  26  furlongs? 

14.  There  are  5760  grains  in  one  pound  Troy ;  how 
many  grains  in  137  pounds  ? 

15.  There  are  256  drams  in  an  ounce ;    how  many 
drams  in  420  ounces  ? 

16.  There   are  198  inches  in   one  rod;    how  manv 
inches  in  76  rods  ? 


(52  MULTIPLICATION. 

17.  There  are  1728  pins  in  a  great  gross ;  how  many 
pins  in  256  great  gross  ? 

18.  There  are  231  cubic  inches  in  a  wine  gallon ;  how 
many  inches  in  48  wine  gallons  ? 

19.  There  are  281  cubic  inches  in  a  beer  gallon;  how 
many  cubic  inches  in  345  beer  gallons  ? 

20.  There  are  4840  square  yards  in  one  acre ;   how 
many  square  yards  in  365  acres  ? 

21.  There  are  5280  feet  in  a  mile ;  how  many  feet  in 
156  miles  ? 

22.  There  are  63360  inches  in  a  mile;    how  many 
inches  in  640  miles  ? 

23.  There  are  5280  feet  in  a  mile;  how  many  feet  in 
the  diameter  of  the  earth,  if  it  is  7912  miles  ? 

PRACTICAL  PROBLEMS 
in  Multiplication. 

1.  How  much  cost  75  barrels  of  flour,  at  7  dollars  a 
barrel  ? 

OPERATION. 
75 

Solution. — If  1  barrel  cost  7  dollars,  75  barrels  - 

will  cost  75  times  7  dollars,  which  are  525  dollars.  — - 

525  Ans. 

Note. — In  practice,  we  multiply  the  75  by  7,  since  it  is  more  con- 
venient to  use  the  smaller  number  as  the  multiplier. 

2.  How  much  will  436  bushels  of  potatoes  cost,  at  48 
cents  a  bushel  ? 

3.  How  much  will  847  bushels  of  corn  cost,  at  56  cents 
a  bushel  ? 

4.  How  much  will  936  yards  of  muslin  cost,  at  37  cents 
a  yard  ? 

5.  A  drover  bought  4896  pigs,  at  9  dollars  each ;  what 
did  they  cost  ? 

6.  How  much  will  3686  grammars  cost,  at  54  cents 
apiece  ? 

7.  At  6  cents  a  quart,  what  will  3678  quarts  of  milk 
cost  ?  9876  quarts  ? 


MULTIPLICATION.  63 


8.  There  are  60  seconds  in  one  minute ;  how  many 
seconds  in  6725  minutes?     In  9360  minutes? 

9.  If  27  men  do  a  piece  of  work  in  48  days,  how  long 
will  it  take  one  man  to  do  it? 

10.  If  145  men  do  a  piece  of  work  in  246  days,  how 
long  at  this  rate  would  it  take  one  man  ? 

11.  If  29  men  build  a  fence  in  276  days,  how  long 
would  it  take  one  man  to  do  it? 

12.  If  200  acres  of  corn  can  be  hoed  by  157  boys  in 
19  days,  how  long  would  it  take  one  boy  ? 

13.  If  35S  men  cut  700  cords  of  wood  in  179  days, 
how  Ionic  would  it  take  one  man  to  do  it  ? 

14.  There  are  16536  letters  in  a  book ;  how  many 
letters  in  496  of  the  same  books  ? 

15.  If  sound  moves  1120  feet  in  one  second,  how  far 
^vill  it  move  in  9872  seconds? 

16.  If  the  earth  moves  in  its  orbit  1640000  miles  in  a 
day,  how  many  miles  does  it  move  in  305  days  ? 

17.  The  moon  is  240000  miles  from  the  earth,  and  the 
sun  about  396  times  as  far ;  how  far  is  the  sun  from  the 
earth  ? 


PRACTICAL   PROBLEMS 
in  Addition,  Subtraction,  and  Multiplication. 

(Quite  young  pupils  may  omit  these  until  review.) 

1.  A  has  245  acres  of  land,  and  B  has  3  times  as  much  ; 
how  many  acres  has  B  ?  how  many  acres  have  both  ? 

2.  One  farmer  has  476  hens,  and  another  farmer  has 
5  times  as  many,  minus  392  hens  j  how  many  has  the 
second  farmer  ? 

3.  One  ship  sailed  1248  miles,  and  another  sailed  8 
times  as  far,  lacking  697  miles ;  how  many  miles  did  the 
second  ship  sail  ? 

4.  A  has  485  dollars,  and  B  has  692  dollars;  how 


64  MULTIPLICATION. 

much  money  have  they  both  ?  how  much  has  C,  if  he 
has  7  times  as  much  as  both  ? 

5.  Mr.  Shank  has  6450  bushels  of  corn,  and  Mr.  Frantz 
has  16  times  as  much,  minus  24986  bushels ;  how  many 
bushels  has  Mr.  Frantz  ? 

6.  A  has  456  dollars,  B  has  759  dollars,  and  C  has  25 
times  as  much  as  both,  minus  8965  dollars ;  how  many 
dollars  have  A  and  B  ?  how  many  has  C  ? 

7.  A  man  sold  24  cows  at  35  dollars  each,  and  17 
horses  at  275  dollars  each ;  what  did  he  receive  for  his 
cows  ?  for  his  horses  ?  for  all  ? 

8.  A  man  sold  his  house  for  4560  dollars,  and  148 
acres  of  land  at  245  dollars  an  acre ;  how  much  did  he 
receive  for  his  house  and  land  ? 

9.  A  bought  126  pigs  at  8  dollars  each,  and  B  bought 

97  sheep  at  12  dollars  each;  which  cost  the  most,  and 
how  much  ? 

10.  B  bought  a  house  for  2960  dollars,  and  gave  for  it 

98  cows  at  24  dollars  each,  and  the  rest  in  money ;  how 
much  money  did  he  pay  ? 

11.  One  army  contains  4575  men,  and  another  36 
times  as  many,  lacking  1936  men ;  how  many  men  in 
the  second  army  ? 

12.  Mr.  Peters  has  2461  gallons  of  coal  oil,  Mr.  Martin 
has  1146  gallons,  and  Mr.  Benson  has  147  times  as  much 
as  both  ;  how  much  has  Mr.  Benson  ? 

13.  A  farmer  sold  129  cows  at  37  dollars  each,  and  re- 
ceived in  payment  2000  dollars ;  how  much  yet  remains 
due  ? 

14.  B  sold  76  hens  at  73  cents  each,  96  turkeys  at  324 
cents  each,  and  received  in  payment  24000  cents;  how 
much  remains  due  ? 

15.  A's  barn  cost  2485  dollars,  his  house  cost  3  times 
as  much,  and  his  farm  cost  as  much  as  both;  what  was 
the  cost  of  the  house  ?  what  was  the  cost  of  the  farm  ? 

16.  A  drover  bought  36  horses  at  145  dollars  a  head, 


DIVISION.  65 

and  96  cows  at  28  dollars  a  head  j  which  cost  the  most, 
and  how  much? 

17.  A's  book  contains  248  pages,  with  2850  letters  on 
a  page,  and  B's  contains  325  pages,  with  3465  letters  on  a 
page ;  how  many  letters  in  A's  book  ?  how  many  in  B's  ? 

18.  A  man  has  75  bags  of  apples,  each  bag  containing 
2  bushels  j  how  much  will  he  receive  for  them,  at  125 
cents  a  bushel  ? 

19.  A  farmer  sold  25  firkins  of  butter,  each  firkin  con- 
taining 126  pounds,  and  received  for  each  pound  37 
cents ;  how  much  did  he  receive  for  it  all  ? 


DIVISION. 


52  Division  is  the  process  of  finding  how  many 
times  one  number  is  contained  in  another. 

53.  The  Dividend  is  the  number  which  contains  the 

other. 

54.  The  Divisor  is  the  number  contained   in  the 

dividend. 

55.  The  Quotient  is  the  number  which  shows  how 
many  times  the  dividend  contains  the  divisor. 

56.  The  sign  of  Division  is  -t-,  and  is  read  divided 
by.  It  shows  that  the  number  on  the  left  is  to  be  divided 
by  the  one  on  the  right. 

57.  There  are  two  methods  of  performing  division 
called  Short  Division  and  Long  Division. 

.     Note  to  Teachers. — If  the  pupils  .are  not  familiar  with  the  ele- 
mentary quotients,  let  them  turn  to  page  21  and  learn  them. 

SHORT    DIVISION. 

58.  Sliort  Division  is  the  method  of  dividing  wher 

the  partial  dividends  are  not  written. 

6* 


06  SHORT   DIVISION. 

Case  I. 
59.  When  tlie  divisor  is  one  figure. 

1.  How  many  times  is  2  contained  in  6  ? 

Solution  1. — We  write  the  6,  draw  a  line  be-         operation. 
neath  and  a  curve  to  the  left,  and  place  the  2  2)6 

to  the  left  of  the  curve.     Two  is  contained  in  6,  3 

three  times,  since   3  times  2  are   6.     We  write 
the  quotient  3  beneath  the  dividend. 

Solution  2. — Two  is  contained  in  6  three  times,  with  no  remainder. 


(2-) 

2)8 

(3.) 

3)6 

2)4 

(5.) 
3)9 

(6.) 
4)8 

(7.) 

2)10 

(8.) 

2)12 

(9.) 
3)12 

(10.) 
2)14 

(11.) 
3)18 

(12.) 
2)20 

(13.) 

2)22 

(14.) 

2)24 

(15.) 
3)21 

(16.) 

3)27 

(17.) 
3)30 

(18.) 
3)36 

(19.) 
3)33 

(20.) 
4)16 

(21.) 
4)24 

(22.) 
4)28 

(23.) 
4)20 

(24.) 
4)36 

(25.) 

4)48 

(26.)  (27.)  (28.)  (29.)  (30.)  (31.) 

5)15  5)25  5)35  5)20  5)40  5)55 

(32.)  (33.)  (34.)  (35.)  (36.)  (37.) 

5)60  6)12  6)24  6)36  6)48  6)60 

(38.)  (39.)  (40.)  (41.)  (42.)  (43.) 

7)21  7)35  7)49  7)63  7)77  7)84 

(44.)  (45.)  (46.)  (47.)  (48.)  (49.) 

8)16  8)64  8)56  8)40  8)72  8)96 

(50.)  (51.)  (52.)  (53.)  (54.)  (55.) 

9)27  9)45  9)63  9)81  9)108  9)99 


SHORT   DIVISION.  67 


Case  I. 

60.  When  the  divisor  is  one  figure  and  there  arp 
no  remainders. 

1.  Divide  46  by  2. 

Solution  1.— 2  is  contained  in  4  tens  2  tens  operation. 

timeo.     2  is  contained  in  6  units  3  units  times  ;  2)46 

hence  the  quotient  is  23.  23 

Solution  2. — 2  is  contained  in  4,  2  times ;  2  is  contained  in  6, 
3  times. 

(2.)  (3.)  (4.)  (5.)  (6.)  (7.) 

2)42         2)48         2)26         2)64         2)84         2)86 

(8.)  (9.)  (10.)  (11.)  (12.)  (13.) 

2)82         3)36         3)69         3)96         3)90         3)39 

(14.)  (15.)  (16.)  (17.)  (18.)  (19.) 

4)48        4)44        4)88        4)40        4)80         5)50 


(20.) 
2)428 

(21.) 
2)228 

(22.) 
2)848 

(23.) 
2)408 

(24.) 
3)369 

(25.) 
3)693 

(26.) 
3)906 

(27.) 
3)609 

(28.) 
3)930 

(29.) 
4)480 

(30.) 

4)804 

(31.) 
4)408 

(32.) 
3)669 

(33.) 
4)488 

(34.) 

2)880 

(35.) 

2)804 

(36.) 
3)906 

(37.) 

2)886 

Case  III. 

(38.) 
2)468 

(39.) 
3)603 

61.  When  the  divisor  is  one 

figure  i 

and  there  are 

remainders. 

1.  Divide  7 

by  3. 

(58  SHORT    DIVISION. 

Solution  1. — Three  is  contained  in  7,  2  times,         operation. 
which  we  write  under  the  7 ;   and  since  2  times  3(7 

3   are  6,  and  7   is   1  more  than  6,  hence  3  is  2  «f- 1 

contained  in  7,  2  times,  and  1  remaining,  which 
we  write  after  the  2  with  the  sign  -j-  before  it. 

Solution  2. — 3  is  contained  in  7,  2  times.    2  times  3  are  6,  6  from 
7  leaves  1 ;  hence  the  quotient  is  2,  with  a  remainder  of  1. 


(2.) 

(3.) 

(4-) 

(5.) 

(6.) 

(7-) 

2)9 

2)11 

2)19 

2)21 

2)13 

"2)25 

(8.)  (9.)  (10.)  (11.)  (12.)  (13.) 

3)5  3)11  3)8  3)14  3)17  3)23 

(14.)  (15.)  (16.)  (17.)  (18.)  (19.) 

4)11  4)17  4)22  4)37  4)43  4)27 

(20.)  (21.)  (22.)  (23.)  (24.)  (25.) 

5)12  5)19  5)28  5)38  5)47  5)58 

(26.)  (27.)  (28.)  (29.)  (30.)  (31.) 

6)15  6)21  6)35  6)51  6)65  6)71 

(32.)  (33.)  (34.)  (35.)  (36.)  (37.) 

7)23  7)29  7)38  7)46  7)58  7)80 

(38.)  (39.)  (40.)  (41.)  (42.)  (43.) 

8)23  8)19  8)28  8)36  8)47  8)93 

(44.)           (45.)  (46.)  (47.)  (48.)  (49.) 

9)31  9)26  9)52  9)61  9)70  9)83 


Case  IV. 

62.  When  the  quotient  contains  several  figures 
and  there  are  successive  remainders. 

1.  Divide  536  by  2. 


SHORT    DIVISION.  69 

Solution  1. — 2  is  contained  in  5  hundreds  2  hun-     operation 
ireds  times,  with  1  hundred  remaining;   1  hundred  2)536 

equals  10  tens,  which,  with  3  tens,  equal  13  tens;  2  o^g 

is  contained  in  13  tens  6  tens  times,  and  1  ten  re- 
maining ;   1  ten  equals  10  units,  which,  with  6  units,  equal  16  units ; 
2  is  contained  in  16  units  8  units  times.     Hence  the  quotient  is  268. 

Solution  2. — 2  is  contained  in  5,  2  times,  and  1  remaining;  2  is 
contained  in  13,  6  times,  and  1  remaining ;  etc. 

Rule — 1.  Write  the  divisor  at  the  left  of  the  dividend; 
begin  at  the  left  hand,  and  divide  the  number  denoted  by 
each  figure  of  the  dividend  by  the  divisor,  and  write  the  quo- 
tient beneath. 

2.  If  there  is  a  remainder  after  any  division,  regard  it 
as  prefixed  to  the  next  figure,  and  divide  as  before.  If  any 
partial  dividend  is  less  than  the  divisor,  prefix  it  to  the  next 
figure,  and  write  a  cipher  in  the  quotient. 

63.  Proof. — Multiply  the  quotient  by  the  divisor, 
and  add  the  remainder,  if  any,  to  the  product. 


(2-) 
2)456 

(3.) 
2)736 

(4.) 

2)548 

(5.) 
2)374 

(6.) 
2)538 

(70 

3)735 

(8.) 
3)816 

(9.) 
3)522 

(10.) 
3)414 

(11.) 

3)738 

(12.) 
3)567 

(13.) 
3)513 

(14.) 
3)645 

(15.) 
3)765 

(16) 
3)825 

(17.) 
4)512 

(18.) 
4)624 

(19.) 

4)732 

(20.) 

4)576 

(21.) 
4^824 

(22.) 
4)736 

(23.) 
4)816 

(24.) 
4)972 

(25.) 

4)608 

(26) 
4)436 

(27.) 
5)615 

(28.) 

5)735 

(29.) 
5)645 

(30.) 

5)785 

(31.) 

5)840 

(32.) 

5)815 

(33.) 

5j935 

(34.) 

5)780 

(85.) 

5)765 

(36.) 

5)980 

70 


► 

] 

LONG    DIVISION. 

(37.) 

(38.) 

(39.) 

(40.) 

(41.) 

6)834 

6)738 

6)654 

6)774 

6)864 

(42.) 

(43.) 

(44.) 

(45.) 

(46.) 

6)1476 

6)3336 

6)2514 

6)3654 

6)7236 

(47.) 

(48.) 

(49.) 

(50.) 

(51.) 

7)2569 

7)4732 

7)8456 

7)9359 

7)9870 

Divide 

Divide 

52.  8256 

by  8. 

59. 

72352  by  8. 

53.  7656 

by  8. 

60. 

23769  by  9. 

54.  9576 

by  8. 

61. 

73145  by  5. 

55.  9874 

by  9. 

62. 

5882597  by 

7. 

56.  9756 

by  9. 

63. 

1101032  by 

8. 

57.  9387 

by  9. 

64. 

21820708  by 

4. 

58.  92565  by  9. 

65. 

6328476  by 

9. 

LONG   DIVISION. 
64.  Long  Division  is  the  method  of  dividing  when 
the  partial  dividends  are  written. 


Case  I. 

65.  When  the  divisor  and  quotient  are  each  one 
figure. 


OPERATION. 

2)7(3 
6 


1.  Divide  7  bv  2. 

Solution  1. — 2  is  contained  in  7  three  times.  We 
place  ihe  3  at  the  right  in  the  quotient,  and  multiply 
the  divisor  by  it.  3  times  2  are  6,  which  we  write 
under  the  7.     We  then  draw  a  line  beneath,  and  sub-  1 

tract,  and  have  1  remaining. 

Solution  2. — 2  is  contained  in  7,  3  times ;  3  times  2  are  6  ;  6  from 
7  leaves  1 ;  hence  the  quotient  is  3,  and  1  remaining. 


(2) 

(3.) 

(*•) 

(5.) 

(6.; 

(?■) 

2)5( 

2)9( 

2)10( 

2)12( 

2)15( 

2)18( 

long  : 

DIVISION 

71 

(8.) 

2)13( 

(9-) 
2)11( 

(10.) 

3)6( 

(11.) 

3)9( 

(12.) 

3)8( 

(13.) 

3)12( 

(14.) 

8)18( 

(15.) 

3)21( 

(16.) 

3)17( 

(17.) 
3)19( 

(18.) 
3)23( 

(19.) 
3)27( 

(20.) 
4)8( 

(21.) 
4)12( 

(22.) 
4)20( 

(23.) 
4)28( 

(24) 
4)30( 

(25) 
4)10( 

(26.) 

4)13( 

(27.) 
4)23( 

(28.) 
4)27( 

(29.) 

5)10( 

(30.) 
4)20( 

(31.) 
5)25( 

(32.) 

6)45( 

(33.) 
5)25( 

(34.) 

5)27( 

(35.) 
5)38( 

(36.) 

5)43( 

(37.) 

5)47( 

(38.) 
6)12( 

(39.) 

6)24( 

(40.) 

6)34( 

(41.) 
6)50( 

(42.) 
6)59( 

(43.) 
6j59( 

(44.) 

7)28( 

(45.) 
7)49( 

(46.) 

7)50( 

(47.) 
7)60( 

(48.) 

7)48( 

(49.) 

7)57( 

(50.) 

8)24( 

(51.) 
8)37( 

(52.) 
8)70( 

(53.) 
8)69( 

(54.) 
8)59( 

(55.) 
8)76( 

(56.) 
9)27( 

(57.) 
9)63( 

(58.) 

9)57( 

(59.) 
9)70( 

(60.) 

9)76( 

(61.) 

9)89( 

Case  II. 

66.   When  the  divisor  is  one  figure  and  the  qno< 
tient  is  several  figures. 

1.  Divide  867  by  3. 

Solution  1. — 3  is  contained  in  8  hundreds  2  operation. 

hundreds   times.     2  hundreds  times  3  equal  6  3)867(289 

hundreds.    6  hundreds  from  8  hundreds  leave  2  6 

hundreds.     2  hundreds  ami  6  tens  are  26  tens.  oq 

3  is  contained  in  26  tens  8  tens  times.     8  tens  24 

times  3  are  24  tens.     24  tens  from  26  tens  leave  ~~^~ 

2  tens.     2  tens  and  7  units  are  27  units.     3  is  27 
contained  in  27  units  9  times.     9  times  3  are  27. 
Subtracting,  nothing  remains.     Hence,  the  quotient  is  287. 


72 


LONG   DIVISION. 


Solution  2. — 3  is  contained  in  8,  2  times  ;  2  times  3  are  6  ;  6  from 
8  leaves  2.  Bring  down  the  6,  and  we  have  26.  3  is  contained  in 
26,  8  times;  8  times  3  are  24 ;  24  from  26  leaves  2.  Bring  down  the 
7.  and  we  have  27.  3  is  contained  in  27,  9  times  ;  9  times  3  are  27, 
etc. 

EXAMPLES   FOR   PRACTICE. 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

2)36(/    2)58( 

2)54( 

2)92( 

2)97( 

(7-) 

(8.) 

(9.) 

(10.) 

(11.) 

3)576( 

3)465( 

3)723( 

3)873( 

3)675( 

(12.) 

(13.) 

(14.) 

(15.) 

(16.) 

4)852( 

4)764( 

4)932( 

4)576( 

4)748( 

(17.) 

(18.) 

(19.) 

(20.) 

(21.) 

5)735( 

5)850( 

5)975( 

5)745( 

5)835( 

(22.) 

(23.) 

(24.) 

(25.) 

(26.) 

6)732( 

6)846( 

6)924( 

6)972( 

6)834( 

(27.) 

(28.) 

(29.) 

(30.) 

(31.) 

7)784( 

7)798( 

7)833( 

7)966( 

7)959( 

(32.) 

(33.) 

(34.) 

(35.) 

(36.) 

8)896( 

8)936(' 

8)9440 

8)976(!n 

8)992( 

Divide 

Divide 

37. 

37596  by  2. 

46. 

46542  by  3. 

38. 

57672  by  3. 

47. 

785641  by  6. 

39. 

78908  by  4. 

48. 

218030  by  8. 

40. 

93546  by  6. 

49. 

51600  by  4. 

41. 

73455  by  5. 

50. 

84507  by  7. 

42. 

75448  by  8. 

51. 

61243  by  2. 

43. 

45794  by  7. 

52. 

47065  by  5. 

44. 

36783  by  9. 

53. 

31696  by  6. 

45. 

487652  by  7. 

54. 

20040  by  9. 

LONG   DIVISION.  73 


Case  III. 
67.  Wlien  the  divisor  is  two  or  more  figures. 

1.  Divide  442  by  13. 

Solution. — 13  is  contained  in  44  tens  3  tens  operation. 

times:,  3  tens  times  13  equal   39    tens;   39  tens  13)442(34 
from  44  tens  leave  5  tens,  and  bringing  down  39 

the  2  units  we  have  52  units.     13  is  contained  in  52 

52  units  4  times.     4  times  13  are  52  ;  subtract-  52 

ing,  nothing  remains.     Hence  the  quotient  is  34. 

Note. — With  young  pupils,  abbreviate  the  explanation,  as  in  the 
previous  solutions. 

Eule. — 1.  Divide  the  number  expressed  by  the  least  num- 
ber of  figures  on  the  left  that  will  contain  the  divisor,  and 
place  the  quotient  on  the  right. 

2.  Multiply  the  divisor  by  this  quotient ;  write  the  product 
under  the  partial  dividend,  and  subtract,  and  to  the  re- 
mainder annex  the  next  figure  of  the  dividend. 

3.  Divide  as  before  until  all  the  figures  of  the  dividend 
have  been  brought  down  ant  used. 

4.  If  any  partial  dividend  will  not  contain  the  divisor, 
place  a  cipher  in  the  quotient,  annex  the  next  figure  of  the 
dividend,  and  proceed  as  before. 

6S.  Proof. — Multiply  the  quotient  by  the  divisor, 
and  add  the  remainder,  if  any,  to  the  product. 

Notes. — 1.  The  pupils  will  notice  that  there  are  four  operations: 
1st,  Divide,  2d,  Multiply,  3d,  Subtract,  4th,  Bring  down. 

2.  If  when  we  multiply  the  product  is  greater  than  the  partial 
dividends,  the  quotient  figure  is  too  large,  and  must  be  diminished. 

3.  When  a  remainder  is  equal  to  or  greater  than  the  divisor,  the 
quotient  figure  is  too  small,  and  must  be  increased. 

4.  A  final  remainder  may  be  set  off  by  itself,  or  it  may  be  written 
over  the  divisor  and  annexed  to  the  quotient. 


74 


LONG  DIVISION. 


EXAMPLES 

Divide 

2.  364  by  11. 

3.  780  by  12. 

4.  312  by  13. 

5.  322  by  14. 

6.  570  by  15. 

7.  752  by  16. 

8.  425  by  17. 

9.  594  by  18. 

10.  608  by  19. 

11.  945  by  21. 

12.  2760  by  22. 

13.  2852  by  23. 

14.  3168  by  24. 

15.  5575  by  25. 

16.  6396  by  26. 

17.  6777  by  27. 

18.  10136  by  28. 

19.  11948  by  29. 

20.  19778  by  31. 

21.  16864  by  32. 

22.  10725  by  33. 

23.  20808  by  34. 

24.  7875  by  35. 

25.  20616  by  36. 

26.  41602  by  37. 

27.  39790  by  38. 

28.  48725  by  39. 

29.  67314  by  41. 

30.  82307  by  42. 

31.  57256  by  43. 

32.  49378  by  44 

33.  98716  by  45. 

34.  60904  by  46. 

35.  76704  by  47. 


FOR  PRACTICE. 

Divide 

36.  62377  by  49. 

37.  84309  by  57. 

38.  92736  by  83. 

39.  41875  by  123, 

40.  106750  by  50. 

41.  120054  by  51, 

42.  116532  by  52. 

43.  133242  by  53. 

44.  126414  by  54. 

45.  132715  by  55. 

46.  146552  by  56. 

47.  154926  by  57, 

48.  126614  by  58. 

49.  191219  by  59. 

50.  234606  by  61. 

51.  259284  by  62. 

52.  274176  by  63. 

53.  301952  by  64. 

54.  234455  by  65. 

55.  186846  by  66. 

56.  423867  by  69. 

57.  478608  by  78. 

58.  C25902  by  87. 

59.  811332  by  372. 

60.  1234560  by  247, 

61.  3456780  by  356. 

62.  1646301  by  381. 

63.  1985175  by  425. 

64.  1787160  by  562. 

65.  2100315  by  581. 

66.  1019806  by  893. 

67.  74818S8  by  1021. 

68.  5226412  by  2036. 

69.  23456789  by  465- 


LONG    DIVISION. 


75 


OPERATION. 

5,00)76,54 

15-154 
or   15i§A 


Case  IY. 
69.   When  ciphers  are  on  the  right  of  the  divisor. 

1.  Divide  7654  by  500. 

Solution". — We  find  how  many  times  5  hun- 
dreds is  contained  in  76  hundreds  by  dividing  76 
by  5.  It  is  contained  15  times,  with  a  remain- 
der of  1  hundred,  which,  with  54,  equals  154. 

Eule. — 1.  Cut  off  the  ciphers  at  the  right  of  the  divisor, 
and  as  many  places  from  the  right  of  the  dividend. 

2.  Divide  the  remaining  part  of  the  dividend  by  the 
remaining  part  of  the  divisor ;  prefix  the  remainder  to  the 
figures  cut  off,  for  the  true  remainder. 

Note. — When  the  divisor,  with  the  ciphers  cut  off,  is  greater  than 
12,  we  will  of  course  divide  by  long  division. 


Divide 

2.  189  by  50. 

3.  487  by  60. 

4.  985  by  80. 

5.  1837  by  400. 

6.  2572  by  1100. 

7.  4783  by  1200. 

8.  8725  by  1300. 

9.  4687  by  1400. 
10.  9876  by  1500. 


Divide 

11.  18732  by  1600. 

12.  28732  by  1700 

13.  19873  by  1900. 

14.  25307  by  2100. 

15.  40302  by  2500. 

16.  87316  by  3400. 

17.  92913  by  4600. 

18.  31200  by  5100. 

19.  8732000  by  12300. 


PRACTICAL  PROBLEMS. 
Case  I. 
TO.  To  divide  a  number  by  an  equal  part. 

1.  At  5  dollars  each,  how  many  sheep  can  you  buy 


for  675  dollars  ? 

Solution. — If  5  dollars  will  buy  one  sheep, 
675  dollars  will  buy  as  many  sheep  as  5  is  con- 
tained times  in  675,  which  are  135.  Hence, 
you  can  buy  135  sheep. 


OPERATION. 

5)675 


135  Ans 


76  PRACTICAL   PROBLEMS. 

2.  At  12  dollars  each,  how  many  pigs  can  you  buy 
for  3780  dollars  ?  Ana.  315. 

3.  At  6  cents  apiece,  how  many  oranges  can  you  buy 
for  354  cents  ?  Ans.  59. 

4.  At  11  cents  a  quart,  how  many  quarts  of  cherries 
can  you  buy  for  1243  cents?  Ans.  113. 

5.  In  one  pound  there  are  12  ounces;  how  many 
pounds  in  1728  ounces  ? 

6.  In  one  minute  there  are  60  seconds ;  how  many 
minutes  in  12900  seconds? 

7.  How  many  cows  can  you  buy  for  2952  dollars,  at 
the  rate  of  24  dollars  each  ? 

8.  How  many  pounds  of  butter  will  8100  cents  buy, 
at  the  rate  of  25  cents  a  pound  ? 

9.  There  are  16  ounces  in  one  pound;  how  many 
pounds  in  5472  ounces  ? 

10.  In  one  bushel  there  are  32  quarts ;  how  many 
bushels  are  there  in  16182  quarts  ? 

11.  How  many  acres  of  land  at  56  dollars  an  acre  can 
you  buy  for  12152  dollars? 

12.  How  long  will  it  take  a  vessel  to  sail  6460  miles, 
at  the  rate  of  68  miles  a  day  ? 

13.  The  diameter  of  the  earth  is  nearly  8000  miles ; 
how  long  will  it  take  a  person  to  walk  the  distance,  at 
the  rate  of  48  miles  a  day  ? 

14.  The  circumference  of  the  earth  is  nearly  25000 
miles ;  how  long  will  it  take  a  person  to  walk  it,  at  the 
rate  of  50  miles  a  day  ? 

15.  The  distance  to  the  moon  is  240,000  miles ;  how 
long  would  it  take  a  balloon  to  reach  it,  moving  at  the 
rate  of  75  miles  an  hour  ? 

16.  The  sun  is  95,000,000  miles  from  the  earth  ;  how 
long  would  it  require  a  cannon-ball  to  reach  it,  moving 
at  the  rate  of  48  miles  a  minute  ? 


PRACTICAL    PROBLEMS.  77 

Case  II. 

Tl.  To  divide  a  number  into  equal  parts. 

1.  A  man  divides  387  dollars  equally  among  9  boys; 
how  many  dollars  does  each  receive  ? 

Solution. — Each  boy  will  receive  as  many         operation. 
dollars  as  9  is  contained  times  in  387,  which  9)387 

are  43  dollars.  43  ^ns. 

2.  A  lady  divides  4860  dollars  equally  among  12  girls; 
how  many  dollars  will  each  receive  ?  Ans.  405. 

3.  A  man  earns  2639  dollars  in  13  weeks ;  how  much 
does  he  earn  in  one  week  ?  Ans.  203. 

4.  A  man  travels  1728  miles  in  36  days  ;  how  far  does 
he  travel  each  day  ? 

5.  There   are   25   pounds  in  a  quarter;   how  many 
pounds  are  there  in  34450  quarters  ? 

6.  There  are  6468  cubic  inches  in  28  gallons;  how 
many  cubic  inches  in  one  gallon  ? 

7.  Sound  moves  37060  feet  in  34  seconds ;  how  far  will 
it  move  in  48  seconds  ? 

8.  There   are   2583   gallons  in   41   hogsheads;    how 
many  gallons  in  one  hogshead? 

9.  If  a  road  57  miles  long  cost  7695  dollars,   how 
much  did  it  cost  a  mile  ? 

10.  A  man  gave  1725  dollars  for  cows  worth  25  dollars 
each  ;  how  many  cows  did  he  buy? 

11.  How  many  bushels  of  oats  at  56  cents  a  bushel 
can  be  bought  for  13272  cents  ? 

12.  A  man  gave  1905  dollars  for  saddles  worth  15 
dollars  each ;  how  many  did  he  buy  ? 

13.  A  farmer   sold  24   horses  for  5640  dollars;    how 
much  did  he  receive  apiece  for  them  ? 

14.  A  farmer  sold  a  lot  of  horses  for  7685  dollars; 
how  many  did  he  sell,  if  he  received  145  dollars  each  ? 

15.  How  many  mules  can  you  buy  for  8832  dollars, 

at  the  rate  of  184  dollars  each  ? 

7* 


78  PRACTICAL   PROBLEMS   IN   ADDITION. 

MISCELLANEOUS  PROBLEMS. 

(Quite  young  pupils  will  omit  these  until  review.) 

PRACTICAL    PROBLEMS    IN  ADDITION. 

1.  A  man  left  850  dollars  to  his  daughter,  and  9-45 
dollars  to  each  of  his  two  sons ;  how  much  did  he  leave 
his  two  sons  ?  how  much  did  he  leave  all  ? 

2.  "Washington  was  born  in  the  year  1732,  Jefferson 
11  years  after,  and  Hamilton  15  years  after  Jefferson; 
when  was  Jefferson  born  ?  when  was  Hamilton  born  ? 

3.  A  farmer  owns  three  farms ;  the  first  is  worth  6560 
dollars,  the  second  385  dollars  more,  and  the  third  1387 
dollars  more  than  the  second  j  what  is  the  value  of  the 
second  farm  ?  of  the  third  farm  ?  of  all  ? 

4.  A  has  7586  cents,  B  has  596  more  than  A,  and  C 
has  as  many  as  A  and  B  together;  how  many  has  B? 
how  many  has  C  ?  how  many  have  all  ? 

5.  B  walked  876  miles,  C  walked  285  miles  more  than 
B,  and  D  walked  985  miles  more  than  C ;  how  far  did 
C  walk  ?  how  far  did  D  walk  ?  how  far  did  they 
together  walk  ? 

6.  A  man  gave  to  his  wife  4675  dollars,  to  his  son  7582 
dollars,  to  his  daughter  3594  dollars,  and  had  8575  dol- 
lars left ;  what  was  his  fortune  ? 

7.  A  owns  a  farm  worth  3750  dollars,  a  wood-lot  worth 
856  dollars  more,  and  a  store  worth  987  dollars  more 
than  both  ;  what  was  the  value  of  the  wood-lot  ?  of  the 
store  ?  of  all  three  ? 

8.  A  man  had  two  sons  and  three  daughters  ;  he  gave 
each  son  5896  dollars,  and  each  daughter  4385  dollars; 
how  much  did  he  give  to  his  sons  ?  to  his  daughters  ? 
to  ail  ? 

9.  A  butcher  sold  to  one  man  876  pounds  of  meat,  to 
another  man    587  pounds   more,  and  to  another,  395 


PRACTICAL   PROBLEMS.  79 

pounds  more  than  both  j  how  much  did  he  sell  to  the 
second  man  ?  to  the  third  man  ?  how  much  to  all  ? 

10.  A  raised  345(3  bushels  of  wheat,  which  was  2475 
bushels  less  than  B  raised,  and  D  raised  3489  bushels 
more  than  both ;  how  much  did  B  raise  ?  how  much 
did  D  raise  ?  how  much  did  all  raise  ? 

11.  A  bought  some  land  for  8759  dollars,  a  house  for 
3768  dollars,  and  sold  them  so  as  to  gain  1389  dollars; 
for  what  did  he  sell  them  ? 

12.  A  man  bought  two  lots  for  3750  dollars  each ;  and 
iD  selling  them  he  gained  278  dollars  on  the  first,  and 
389  dollars  on  the  second;  how  much  did  he  gain  on 
both  ?  how  much  did  he  receive  for  both  ? 

13.  A  has  757  acres  of  land,  B  has  285  acres  more  than 
A,  and  C  has  as  many  as  A  and  B  both ;  how  many 
acres  has  B  ?  how  many  has  C  ?  how  many  have  all  ? 

14.  William  lends  his  brother  3785  dollars,  his  sister 
4261  dollars,  and  a  friend  485  dollars  more  than  his 
sister,  and  has  5858  dollars  remaining;  how  much  did 
he  lend  his  friend,  and  what  was  his  whole  fortune  ? 

PRACTICAL   PROBLEMS 
in  Addition  and  Subtraction. 

1.  Find  the  sum  of  six  hundred  and  five  and  18  hun- 
dred and  ninety-seven. 

2.  Subtract  one  thousand  and  nine  from  four  thousand 
and  seven. 

3.  Subtract  7567  +  896  from  4875  +  4736  -j-  2539. 

4.  A  had  472  hens,  and  bought  589  hens,  and  then 
sold  985  ;  how  many  had  he  then  ? 

5.  A  farmer  had  397  pigs,  and  bought  85  pigs,  and 
then  sold  182  pigs  ;  how  many  had  he  then  ? 

6.  A  drover  sold  his  cows  for  257-")  dollars,  and  his 
sheep  for  976  dollars,  and  gained  594  dollars  ;  how  much 
did  he  pay  for  them  ? 

7.  A  man  having  1600  acres  of  land  sold  546  acres 


80  PRACTICAL   PROBLEMS. 

to  B,  and  289  acres  more  to  C  than  to  B ;  how  much 
did  he  sell  to  C  ?  how  much  to  both  ?  how  much  re- 
mained ? 

8.  Mr.  Peters,  having  4300  bushels  of  wheat,  sold  1480 
bushels,  and  then  bought  1856  bushels  more  than  he 
sold ;  how  many  bushels  had  he  then  ? 

9.  Henry  had  756  dollars,  and  his  mother  gave  him 
enough  to  make  his  money  1200  dollars ;  how  much  did 
his  mother  give  him  ? 

10.  Sarah  bought  575  pins ;  her  mother  gave  her  289, 
and  her  sister  gave  her  enough  to  make  her  number 
1000  j  how  many  did  she  receive  from  her  sister  ? 

11.  Mary's  father  left  her  596  acres  of  land;  she  sold 
484  acres,  and  then  bought  396  acres ;  how  many  acres 
had  she  then  ? 

12.  A  sold  4760  bushels  of  grain,  then  sold  1780 
bushels,  and  then  had  1875  bushels ;  how  many  bushels 
had  he  at  first  ? 

13.  B  sold  7560  bushels  of  rye,  then  bought  2580 
bushels,  and  then  had  5680  bushels ;  how  much  did  he  sell 
more  than  he  bought  ?  how  many  bushels  had  he  at  first  ? 

14.  William  had  456  dollars  ;  his  father  gave  him  2528 
dollars,  he  then  lost  1869  dollars,  and  gave  away  286 
dollars  ;  how  many  dollars  had  he  then  ? 

15.  Three  men  bought  a  farm  for  20000  dollars ;  the 
first  paid  7580  dollars,  the  second  paid  6765  dollars,  and 
the  third  the  remainder ;  how  much  did  the  first  and 
second  pay  ?  how  much  did  the  third  pay  ? 

16.  A  man  deposited  8000  dollars  in  the  bank;  he 
drew  out  at  one  time  2575  dollars,  at  another  3467  dol- 
lars, at  another  1576  dollars ;  how  much  remained  in 
the  bank  ? 

17.  Mr.  Bowman,  whose  property  was  35000  dollars, 
willed  9650  dollars  to  each  of  his  two  sons,  8750  to  his 
daughter,  and  the  remainder  to  his  wife;  how  much  did 
the  children  receive?  how  much  did  his  wife  receive  ? 


PRACTICAL   PROBLEMS.-  81 

PRACTICAL   PROBLEMS 
in  Addition,  Subtraction,  and  Multiplication. 

1.  What  is  the  value  of  467  X  672  —  31675  ? 

2.  What  is  the  value  of  672  X  36  plus  216  X  42  ? 

3.  A  man  sold  his  house  for  27  times  98  dollars,  plus 
397  dollars ;  how  much  did  he  receive  for  it  ? 

4.  A  sold  24  horses  for  168  dollars  each,  and  63  cows 
for  34  dollars  each ;  what  did  he  receive  for  the  horses  ? 
for  the  cows  ?  for  all  ? 

5.  I  bought  76  oxen  at  68  dollars  each,  and  327  sheep 
at  12  dollars  each ;  how  much  did  the  oxen  cost  ?  how 
much  the  sheep  ?  how  much  did  all  cost  ? 

6.  My  barn  cost  2318  dollars,  my  house  3  times  as 
much,  and  my  farm  as  much  as  both ;  what  was  the 
cost  of  the  house  ?  the  cost  of  the  farm  ? 

7.  A  man  bought  235  cows  at  24  dollars  each,  and 
sold  them  for  32  dollars,  each ;  how  much  did  he  gain 
by  the  transaction  ? 

8.  A  drover  bought  78  horses  at  164  dollars  each, 
215  oxen  at  59  dollars  each,  and  sold  them  all  for  30000 
dollars  ;  how  much  did  he  gain  ? 

9.  How  much  must  I  pay  for  7  building-lots,  at  2348 
dollars  each,  5  houses,  at  4250  dollars  each,  and  6  boats, 
at  3980  dollars  each  ? 

10.  I  bought  78  sheep  at  7  dollars  a  head,  and  sold 
them  so  as  to  gain  267  dollars ;  how  much  did  I  receive 
for  them  ? 

11.  I  bought  185  acres  of  land  at  95  dollars  an  acre, 
and  in  selling  it  I  lost  2486  dollars ;  how  much  did  I 
receive  for  it  ? 

12.  A  speculator  bought  a  farm  of  327  acres  at  79 
dollars  an  acre,  and  sold  it  at  95  dollars  an  acre ;  how 
much  did  he  gain  ? 

13.  A  man  sold  his  oil  stock  for  14000  dollars,  and 
then  bought  a  farm  containing  93  acres,  at  125  dollars 


82  PROBLEMS. 

an  acre ;    how   much  money  has  he  left  after  paying 
for  it  ? 

14.  A  clerk  receives  a  salary  of  75  dollars  a  month ; 
he  spends  18  dollars  a  month  for  "board,  and  9  dollars 
for  other  expenses ;  how  much  can  he  save  in  1  month  ? 
in  12  months  ? 

15.  A  farmer  having  3420  dollars  bought  35  cows  at 
24  dollars  a  head,  and  36  oxen  at  54  dollars  a  head; 
how  much  has  he  left,  after  paying  for  them? 

16.  Thomas  travels  24  miles  a  day,  and  Walton  travels 
52  miles  a  day ;  how  much  farther  does  Walton  travel 
in  72  days  than  Thomas  ? 

17.  A  man  bought  336  bushels  of  potatoes  at  65  cents 
a  bushel,  and  3  times  as  many  bushels  of  apples  at  98 
cents  a  bushel ;  what  was  the  entire  cost  ? 

18.  A's  barn  cost  1980  dollars,  his  house  2150  dollars 
more  than  the  barn,  and  his  farm  cost  14  times  as  much 
as  the  barn  and  house  together;  what  was  the  cost  of 
the  farm  ? 

PRACTICAL   PROBLEMS 
in  Addition,  Subtraction,  Multiplication,  and  Division. 

1.  Divide  42624  by  36  and  add  3146  to  the  quotient. 

2.  Divide  73305  by  45  and  subtract  the  quotient  from 
3702. 

3.  Subtract  3125  from  5213,  divide  the  remainder  by 
9,  and  add  the  quotient  to  1745. 

4.  A  man  having  18000  dollars  leaves  his  wife  4800 
and  divides  the  remainder  equally  among  6  children ; 
what  does  each  receive? 

5.  A  farm  of  24  acres  was  bought  for  4056  dollars  and 
sold  at  a  gain  of  3168  dollars :  for  what  was  it  sold  per 
acre  ? 

6.  If  27  men  share  11286  dollars  equally,  how  much 
would  each  have  ?  how  much  would  A  have,  if  he  had 
four  times  as  much  as  each,  plus  1245  dollars? 


PROBLEMS.  83 

7.  If  29  men  earn  7946  cents  in  a  day,  and  25  boys 
earn  5450  cents  in  a  day,  how  much  more  does  one  man 
earn  in  a  day  than  one  boy  ? 

8.  A  horse  and  18  oxen  are  worth  1001  dollars;  now, 
if  the  horse  is  worth  245  dollars,  what  is  the  value  of 
the  oxen  ?  of  each  ox  ?  Ans.  42  dollars. 

9.  The  value  of  3  horses  and  15  cows  is  V 55  dollars; 
if  the  value  of  each  horse  is  225  dollars,  what  is  the 
ralue  of  each  cow  ?       -  Ans.  32  dollars. 

10.  If  you  divide  G0466  by  49,  by  what  number  must 
I  multiply  the  quotient  to  produce  9872  ?  Ans.  8. 

11.  The  income  of  a  man  who  "  struck  oil"  is  400 
dollars  per  day  ;  how  many  teachers  would  this  employ 
at  a  salary  of  730  dollars  a  year  ? 

12.  I  bought  326  barrels  of  flour  for  2608  dollars,  pad 
46  dollars  for  transportation,  and  sold  it  at  a  gain  cf 
280  dollars ;  what  did  I  receive  a  barrel  ? 

Ans.  9  dollars. 

13.  I  sold  a  farm  containing  190  acres  for  65  dollars 
an  acre,  and  bought  with  the  proceeds  another  farm  at 
95  dollars  an  acre ;  how  many  acres  in  the  latter  farm  ? 

14.  A  drover  bought  234  cows  at  25  dollars  each,  and 
sold  95  of  them  at  cost  each ;  how  much  must  he  re- 
ceive a  head  for  the  remainder,  to  gain  973  dollars? 

Ans.  32. 

15.  If  the  President  of  the  United  States  expends  52 
dollars  daily,  how  much  can  he  save  in  a  year  of  365 
«iays,  out  of  his  salary  of  25000  dollars? 

16.  If  the  Vice-President  expends  35  dollars  daily, 
how  much  can  he  save  at  the  end  of  the  year,  if  he  has 
an  income  of  6450  dollars,  besides  his  salary  of  8000 
dollars  a  year  ? 

17.  If  the  Secretary  of  State  expends  16  dollars  a 
day,  how  much  can  he  save  in  a  year,  his  salary  being 
8000  dollars  a  year  and  his  private  income  28  dollars  a 
week  ? 


84  UNITED    STATES    MONEY. 

SECTION    IV. 

UNITED   STATES   MONEY. 

72.    United  States  Money  is  the  money  of  the  United 

States. 

TABLE. 

10  mills  (m.)  equal     ....     1  cent,     c. 

10  cents  " 1  dime,    d. 

10  dimes  " 1  dollar,  $. 

10  dollars  " 1  eagle,  E. 

Coins  are  pieces  of  metal,  stamped  by  the  authority 
of  the  government,  to  be  used  as  money. 

GOLD    COINS.  SILVER    COINS. 


Eagle,                value  $10 

Double-eagle,       •'  20 

Half -eagle,           "  5 

Quarter-eagle,      "  2\ 

Dollar,                  "  1 

Three-dollar,        "  3 


Dollar,  value  $1 

Half-dollar,  "  50  c. 

Quarter-dollar,      "  25  c. 

Dime,  "  10  c. 

Half-dime,  "  5  c. 

Three-cent,  "  5  c. 


NICKEL.  BRONZE. 


Three-cent,  value  3  c. 

Five-cent,        "  5  c. 


Two-cent,  value         2  c. 

Cent,  "  ic. 


NUMERATION  AND  NOTATION. 

73.  The  doZtar  is  indicated  by  the  symbol  $.  The 
eagle  and  dollar  are  read  as  a  number  of  dollars :  thus, 
3  eagles  and  5  dollars  are  read,  35  dollars. 

74.  The  dime  is  one-tenth  of  a  dollar,  and  is  written 
to  the  right  of  the  dollar  and  separated  from  it  by  a 
point,  called  a  separatrix;  thus,  $3.4  represents  3  dol- 
lars and  4  dimes,  ta  '  ! . 

75.  The  cent  is  1  tenth  of  a  dime,  or  1  hundrbdth  of  a 
dollar.  It  is  written  two  places  to  the  right  of  dollars; 
thus,  $4.58  represents  4  dollars,  5  dimes,  and  8  cents. 


UNITED    STATES    MONEY.  85 

76.  Dimes  and  cents  are  usually  read  as  so  many 
cents ;  thus,  $7.45  is  read,  7  dollars  and  45  cents. 

77.  The  mill  is  1  tenth  of  a  cent,  and  is  written  one 
place  to  the  right  of  cents  •  thus,  $5,475  is  read,  5  dollars, 
47  cents,  and  5  mills. 

PRACTICAL   PROBLEMS. 

EXAMPLES    IN    NUMERATION. 

1.  Write  and  read  $24.75. 

Solution. — The  pupil  will  write  this  upon  the  elate  or  black- 
board, and  say:  This  is  read,  24  dollars,  7  dimes,  and  5  cents;  or, 
24  dollars  and  75  cents. 

The  pupil  will  write  and  read  the  following : 


2.  $14.25 

3.  $24.67 

4.  $19.84 

5.  $28,574 


6.  $48.50 

7.  $50.06 

8.  $48,408 

9.  $96,004 


10    $105  076 

11.  $976,705 

12.  $350,035 

13.  $847,008 


EXAMPLES    IN    NOTATION. 

1.  "Write  six  dollars  and  twenty-five  cents. 

2.  Write  twenty-five  dollars  and  thirty-six  cents. 

3.  Write  eight  dollars,  forty-five  cents,  and  six  mills. 

4.  Write  twenty  dollars,  seventy-five  cents,  and  two 
mills. 

5.  Write  six  eagles,  seven  dollars,  and  eighty-four 
cents. 

6.  Write  four  dollars,  six  dimes,  and  seven  cents. 

7.  Write  25  dollars,  five  cents,  and  eight  mills. 

REDUCTION    OF   UNITED    STATES    MONEY. 

78.  Reduction  consists  in  changing  the  denomina~ 
tion  without  changing  the  value.  From  the  table  we 
derive  the  following  principles  : 

79.  To  reduce  cents  to  mills,  we  multiply  the  cents  by  10, 
or  annex  one  cipher. 


66  UNITED    STATES    MONEY. 

80.  To  reduce  dollars  to  cents,  we  annex  two  ciphers. 

81.  To  reduce  dollars  to  mills,  we  annex  three  ciphers. 

82.  To  reduce  a  number  of  dollars  and  cents  to  cents, 
we  remove  the  decimal  point ;  thus,  $5.24  =  524  cents. 

Case  I. 
To  reduce  to  lower  terms. 

1.  Eeduce  6  dollars  to  cents. 

Solution. — In  1  dollar  there  are  100  cents  ;         operation. 
hence,  in  6  dollars  there  are  6  times  100  cents,     $6  =  000  cents- 
or  600  cents ;  or  we  annex  two  ciphers. 

2.  Eeduce  $18  to  cents. 

3.  Eeduce  $24  to  cents. 

4.  Eeduce  $385  to  cents. 

5.  Eeduce  $27  to  mills. 

6.  Eeduce  85  cents  to  mills. 

7.  Eeduce  $5.47  to  cents. 

8.  Eeduce  $27.05  to  cents.  < 

9.  Change  $9  607  to  mills. 

Case  II. 
To  reduce  to  higlier  terms. 

S3.  From  the  table  we  have  the  following  principles  : 

1.  To  reduce  cents  to  dollars,  place  the  point  two  places 
from  the  right. 

2.  To   reduce   mills   to  dollars,  place  the  point  three 
places  from  the  right. 

1.  Eeduce  2347  cents  to  dollars. 

Solution. — There  are  100  cents  in  1  dollar,  operation. 
and  in  2347  cents  there  are  as  many  dollars  as  2347  -f- 100 
100   is    contained    times    in    2347,    which    are  =$23.47 

$23.47  ;   or  we  place  the  point  two  places  from 
the  right. 

2.  Eeduce  845  cents  to  dollars.  Ans.  $8.45. 

3.  Eeduce  2835  cents  to  dollars.  Ans.  $28.35. 


UNITED    STATES    MONEY.  87 

4.  Reduce  4G785  cents  to  dollars. 

5.  Eeduce  7895  mills  to  dollars. 

6.  Eeduce  27005  mills  to  dollars. 

7.  Eeduce  4800  cents  to  dollars. 

8.  Eeduce  9600  mills  to  dollars. 

ADDITION  OF  UNITED  STATES  MONEY. 
84.  Addition  of  United  States  Money  is  performed 
as  in  simple  numbers,  according  to  the  following 

Eule. — 1.  Write  dollars  under  dollars,  cents  under 
cents,  etc. 

2.  Add  as  in  simple  numbers,  and  place  the  separatrix 
between  dollars  and  cents. 

1.  Find  the  sum  of  824.30,  890.58,  and  875.42. 

Solution. — We  -write  dollars   under   dollars  operation. 

and  cents  under  cents,  and   commence  at   the  $24.86 

right  to  add.     2  and  8  are  10,  and  6  are  16  cents ;  96.58 

which  equals  6  cents  and  1  dime;   we  -write  the  75.42 

6  cents  under  the  column  of  cents,  and  add  the  QiQf  on 
1  dime  to  the  next  column,  etc. 

2.  Find  the  sum  of  648.50,  839.46,  824.G7,  and  881.09. 

3.  Add  823.84,  897.30,  852.75,  and  898.27. 

4.  Add  873.75,  848.50,  839.87,  and  875.48. 

5.  Add  840.375,  897.283,  872.475,  and  88.390. 
0.  Add  8150.90,  8284.070,  89.27,  and  885.735. 

7.  A  man  bought  a  cow  for  824.75,  a  horse  for  8150.50, 
a  wagon  for  8287.75,  and  a  carriage  for  8375.87 ;  how 
much  did  he  pay  for  all  ? 

8.  A  merchant  bought  flour  for  857.35,  some  calicc 
for  896.87,  some  cloth  for  884.50,  some  boots  for  852.87, 
and  some  muslin  for  875.75;  what  did  they  all  cost? 

9.  A  tailor  sold  a  coat  for  $34.75,  a  vest  for  88.50, 
a  cloak  for  852.25,  a  pair  of  pants  for  89.75,  and  some 
other  things  for  828.45;  what  did  he  receive  for  all? 

10.  I  bought  a  table  for  818.25,  a  looking-glass  for 


S8  UNITED    STATES    MONEY. 

$25.75,  a  bedstead  for  $36.50,  a  bureau  for  $46.25 ;  wnat 
did  they  all  cost  ? 

11.  A  owes  $624.30,  B  owes  $467.56,  C  owes  $359.45, 
D  owes  $95.12,  E  owes  $43.84,  F  owes  $27.75,  G 
owes  $968.47,  H  owes  $7.75 ;  required  the  sum  of 
their  debts. 

SUBTRACTION  OF  UNITED  STATES  MONEY. 

85.  Subtraction  of  United  States  3Ioney  is  per- 
formed as  in  subtraction  of  simple  numbers,  according 
to  the  following 

Eule. — 1.  Write  dollars  under  dollars,  cents  under 
cents,  etc. 

2.  Subtract  as  in  simple  numbers,  and  place  the  separatrix 
between  dollars  and  cents. 

1.  Subtract  $21.48  from  $46.73. 

Solution. — We  cannot  subtract  8  cents  from         operation. 
6  cents,   hence  we    add    10  cents   to   8   cents,  $46.73 

making  13  cents;  8  cents  from  13  cents  leave  27.48 

5  cents.     Now,  since  we  added  10  cents,  or  1  $19.25 

dime,  to  the  minuend,  we  must  add  1  dime  to 
the  4  dimes,  making  5  dimes  :  5  dimes  from  7  dimes  leave  2  dimes,  etc. 

(2.)  (3.)  (4.)  (5.) 

$78.25  $57.52  $96.43  $75.75 

13.16  23.28  28.14  23.28 


6.  From  $129.39  take  $48.91. 

7.  Find  the  difference  between  $234.16  and  $471.24. 

8.  A  man  bought  a  horse  for  $234.50,  and  sold  it  for 
$228.25  ;  what  did  he  lose  ? 

9.  A  merchant  bought  cloth  for  $96.75,  and  sold  it  for 
6110.29  ;  what  did  he  gain  ? 

10.  A  bought  a  farm  for  $3640.25,  and  sold  it  for 
64000  ?  what  was  the  gain  ? 

11.  My  house  cost  $3480.75,  and  I  sold  it  for  $4000.50; 
what  did  I  gain  ? 


UNITED    STATES    MONEY.  89 

12.  My  horse  cost  $240.50,  and  my  carriage  cost 
$386.25  ;  I  sold  them  for  $680.50  ;  what  did  I  gain  ? 

13.  A  merchant  bought  cloth  for  $325.50,  muslin  for 
$436.75,  and  flour  for  8625.80;  he  sold  them  all  for  $1300; 
how  much  did  he  lose  ? 

14.  I  paid  $4637.25  for  a  farm,  paid  $3675.25  foi 
building  a  house,  and  $2896.87  for  building  a  barn ;  I 
sold  my  property  for  $13000 ;  how  much  did  I  gain  ? 

15.  I  paid  $246.75  for  a  horse,  $325.45  for  a  mule, 
$42.25  for  an  ox,  $37.50  for  a  cow;  I  sold  them  all  for 
$603.50  j  what  was  the  loss  ? 

MULTIPLICATION  OF  UNITED  STATES  MONEY. 

86.  Multiplication  of  United  States  Money  is  per- 
formed, like  multiplication  of  simple  numbers,  according 
to  the  following: 


o 


Rule. — Multiply  as  in  simple  numbers,  and  place  the 
separatrix  between  dollars  and  cents. 

1.  Multiply  $36.25  by  3. 

Solution. — Three  times  5  cents  are  15  cents,  operation. 

which  equal  1    dime  and  5  cents;  we  write  the  $36.25 

5  cents,  and  reserve  the  1  dime  to  add  to  the  next  3 

product.     3  times  2  dimes  are  6  dimes,   and  6  £108  75 
dimes  plus  1  dime  are  7  dimes,  etc. 


Multiply 

2.  $26.14  by  4. 

3.  $37.27  by  5. 

4.  $48.96  by  7. 

5.  837.52  by  8. 

6.  $79.35  by  9. 


Multiply 

7.  $48.25  by  12. 

8.  $72.27  by  13. 

9.  $85.58  by  15. 

10.  $92.83  by  32. 

11.  $75.32  by  46. 


12.  If  one  yard  of  cloth  cost  $3.25,  what  cost  5  yards? 

13.  What  will  12  horses  cost  at  the  rate  of  $150.75 
a  piece  ? 

14.  A  man  bought  27  oxen  at  the  rate  of  $36.25  each  , 
what  did  they  cost  ? 


8* 


90  UNITED    STATES    MONEY. 

15.  A  farmer  sold  325  bushels  of  wheat  at  $1  25  a 
bushel ;  how  much  did  he  receive  for  it  ? 

16.  A  miller  sold  472  barrels  of  flour  at  $7.87  a  barrel; 
how  much  did  he  receive  for  it  ? 

17.  A  man  bought  47  cows  for  $24.30  each,  and  sold 
them  for  $28.10  each ;  what  was  the  gain  ? 

18.  A  drover  bought  247  horses  for  $130.75  each,  and 
gold  them  for  $180.30  each;  what  did  he  gain? 

19.  A  farmer  bought  327  acres  of  land  at  $76.25  an 
acre,  and  sold  it  at  $92.50  an  acre;  what  did  he  gain? 

DIVISION  OF  UNITED  STATES  MONEY. 

87.  Division  of  United  States  Money  is  performed 
like  division  of  simple  numbers. 

Case  I. 

88.  To  divide  a  number  irato  equal  parts. 

Rule. — Divide  as  in  simple  numbers,  and  -place  the  sepa- 
ratrix  between  dollars  and  cents. 

1.  Divide  $7.32  in  3  equal  parts,  or  find  1  third  of  it. 

Solution. — 1  third  of  7  dollars  is  2  dollars,         operation. 
and  1  dollar  remaining;  1  dollar  equals  10  dimes,  3)$7.32 

which,  added  to  3  dimes,  equal    13    dimes.     1  $2.44  Ans. 

third  of  13  dimes  equals  4  dimes,  and  1  dime 
remaining,  etc. 

2.  Divide  $9.24  into  4  equal  parts. 

3.  Divide  $7.25  into  5  equal  parts. 

4.  Divide  $17.22  into  6  equal  parts. 

5.  If  7  pigs  cost  $36.75,  what  will  one  pig  cost  ? 

6.-  If  8  cows  cost  $172.80,  what  will  one  cow  cost? 

7.  If  3  oxen  cost  $325.20,  what  will  5  oxen  cost? 

8.  If  7  hens  cost  $3.15,  what  will  12  hens  cost? 

9.  What  cost  15  sheep,  if  4  sheep  cost  $29.24? 

10.  What  cost  25  pounds  of  butter,  if  7  pounds  cost 
62.38  ? 


UNITED    STATES    MONEY.  91 

11.  What  cost  34  acres  of  land,  if  12  acres  cost  $5.04? 

12.  What  cost  28  cows,  if  35  cows  cost  987  dollars? 

13.  What  cost  75  oxen,  if  38  oxen  cost  1615  dollars? 

14.  What  cost  234  hens,  if  75  hens  cost  $25.50  ? 

Case  II. 

89.  To  divide  one  sum  of  money  by  another, 

Eule. — Reduce  both  sums  to  the  same  denomination,  and 
divide  as  in  simple  numbers. 

1.  Divide  $736  by  $4. 

OPERATION. 

Solution. — Dividing  as  in  simple  numbers,  4)736 

we  have  184.  ~Jj^  Ans# 

2.  Divide  $9600  by  $16. 

3.  Divide  728  cents  by  4  cents. 

4.  Divide  3625  cents  by  5  cents. 

5.  Divide  $26325  by  81  dollars. 

6.  At  24  dollars  each,  how  many  cows  can  you  buy 
for  1344  dollars? 

7.  At  42  dollars  each,  how  many  oxen  can  be  bought 
for  $3276  ? 

8.  At  $3.25  apiece,  how  many  pigs  can  you  buy  for 
$120.25? 

9.  A  earned  $3.75  a  day ;  how  many  days  did  he  work 
to  earn  $78.75  ? 

10.  A  drover  paid  $6972  for  horses,  at  $145.25  apiece; 
how  many  did  he  buy  ? 

11.  How  many  cords  of  wood  can  you  buy  for  $312, 
at  $3.25  a  cord  ? 

12.  William  earned  $3.25  a  day,  and  paid  75  cents  for 
board;  in  how  many  days  would  he  save  S912.50? 

13.  A  merchant  received  $853.25  for  a  case  of  silk, 
including  $1.25  cost  of  box.  How  many  pieces  of  silk 
were  in  the  case,  if  he  received  $53.25  apiece  ? 


92 


BILLS    AND    ACCOUNTS. 


BILLS  AND  ACCOUNTS. 

90.  A  Bill  is  a  written  statement  of  goods  bought 
and  sold,  the  quantity,  price,  and  entire  cost. 

91.  An  Account  is  a  bill  in  which  each  of  the  parties 
has  received  value  of  the  other. 

92.  The  party  who  owes  is  the  debtor;  the  party 
who  is  owed  is  the  creditor.  A  bill  is  made  out  by  the 
following 

Kule.— Multiply  the  cost  of  each  article  by  the  whole 
number,  and  find  the  sum  of  the  products. 

92J.  In  an  account,  find  the  difference  between  the  debit 
and  credit  amounts. 

Make  out  the  following  bills  : 

(1.)  Millersville,  May  8,  186%. 

Mr.  Harry  Bowman, 

Bought  of  HENR  Y  MARTIN, 


8 

yds. 

of 

muslin, 

at 

12 

a 

of 

cloth, 

a 

15 

a 

of 

silk, 

a 

27, 
2.37, 
l.t 


Amount  due, 


(2-) 
Theo.  Miller, 


Lancaster,  April  6,  1864. 
Bought  of  DANIEL  MO  ONE  Y, 


24- 
37 

¥> 
28 


b*=*=s 


o 


©=*=*& 


pairs  boots,  at  $5.25, 
gaiters,  "  3.75, 
slippers,  "  1.37, 
rubbers,    "     1.25, 


a 


u 


Amount  due, 

Received  Payment, 

Theo.  Miller. 


BILLS   AND   ACCOUNTS. 


93 


(3.) 
John  J.  Brooks, 


New  York,  Dec.  17,  1862. 
Bought  of  CHARLES  HOYT, 


47 

28 

97 

U6 


€>SS6^ 


bbls.  of  flour,  at  $7.35, 
lbs.  of  beef,  "  0.37, 
yds.  of  cloth,  "  2.75, 
bu.   of  wheat, "     1.12, 


Amount, 
Received  Payment, 

Charles  Hoyt. 


(4-) 
John  Smith,  Dr. 


1S66. 

Jan. 

1 

Feb. 

5 

Jan. 

7 

Feb. 

2 

To  75  lbs.  of  sugar,  at  $0.35, 
"    4.7  yds.  of  cloth,  "     3.25, 

Cr. 

By  75  bu.  of  corn,  at  $0.78, 
"  83  bu.  of  apples, "     1.25, 

Balance  due, 


$ 

f 

(5.)  Philadelphia,  April  1,  1860. 

Mr.  Henry  Farnam,  Dr. 

To  Edwin  Lamborn. 


I860. 

Jan. 

4. 

Jan. 

10. 

Jan. 

20. 

I860. 

Jan. 

3. 

Jan. 

12. 

Feb. 

21^. 

To  145  bu.  wheat,  at  $1.25, 
"  236  "  rye,  «  1.05, 
"  176  "    oats,     "     0.65, 

Cr. 
By  45  yds.  cloth,        at  $3.65, 
"    72    "    silk,  "     2.12, 

"    £#    "     cassimere, "    i.75, 

Bed  a  nee  due, 
Received  Payment, 

Edwin  Lamborn. 


94  COMMON    FRACTIONS. 


SECTION  V. 
COMMON  FEACTIONS. 

93.  A  Fraction  is  a  number  of  equal  parts  of  a 
unit ;  as  one  half,  two  thirds,  etc. 

94.  A  fraction  is  expressed  by  figures  with  a  line 
between ;  thus,  §  expresses  2  thirds. 

95.  The  number  denoted  by  the  figure  below  the 
line  is  called  the  denominator ;  it  shows  the  number  of 
equal  parts  into  which  the  unit  is  divided. 

98.  The  number  denoted  by  the  figure  above  the  line 
is  the  numerator;  it  shows  the  number  of  equal  parts 
considered. 

97.  A  Proper  Fraction  is  one  whose  value  is  less 
than  a  unit ;  as  §,  f ,  |5  etc. 

98.  An  Improper  Fraction  is  one  whose  value  is 
equal  to  or  greater  than  a  unit;  as  |,  |,  2Bl,  etc. 

99.  A  Compound  Fraction  is  a  fraction  of  a  fraction ; 
as  \  of  |. 

100.  A  Mixed  Number  consists  of  a  whole  number 
and  a  fraction;  as  2 J,  5'f,  etc. 

To  Teachers. — Give  pupils  a  clear  idea  of  a  fraction  by  dividing 
some  object,  as  an  apple,  by  lines  upon  the  blackboard,  etc.  For 
Mental  Exercises,  see  Primary  Mental  Arithmetic. 

MENTAL   EXERCISES. 

1.  What  is  one-half? 
Ans.  One-half  of  any  thing  is  one  of  the  two  equal  part,-  of  it 

What  is  What  is 


2.  One-third? 

3.  One-fourth  ? 

4.  One-fifth? 

5.  One-sixth  ? 


6.  One-seventh  ? 

7.  One-eighth  ? 

8.  One-tenth  ? 

9.  One-twelfth? 


COMMON    FRACTIONS. 


95 


1.  What  is  two-thirds? 
Ans.  Two-thirds  of  any  thing  is  two  of  the  three  equal  parts  of  it. 

What  is  What  is 


2.  Two-fourths? 

3.  Three-fourths? 

4.  Two-fifths? 

5.  Three-fifths? 


6.  Four-fifths? 

7.  Two-sixths? 

8.  Three-sevenths? 

9.  Four-ninths  ? 


1.  What  is  J  of  6  ? 

Ans.  \  of  6  is  3,  since  2  times  3  are  6. 


2. 

Find  I  of  8. 

5. 

Find  |  of  15. 

o 
O. 

Find  |  of  12. 

6. 

Find  |  of  20. 

4. 

Find  \  of  16. 

7. 

Find  4  of  30. 

0 

1. 

2. 


NUMERATION  AND  NOTATION. 
Eead  the  following  fractions. 

2     3  .   _6_ 
°'    5  >    ll' 

A     11.      8 

^'   T3  >    10"" 


5  .  6 

G>  7" 

7  .  3 

3  >  5 


5. 
6. 


i  f? . 

•2  i   > 


7 
To' 


53-     115 


Write  the  following:  fractions. 


1.  Two-thirds. 

2.  Four-fifths. 

3.  Five-sevenths. 

1.  Analyze  the  fraction  f. 


4.  Eight-tenths. 

5.  Seven-ninths. 

6.  Eleven-fifteenths. 


Solution. — In  the  fraction  f,  the  denominator,  4,  shows  that  the 
unit  is  divided  into  4  equal  parts,  and  the  numerator,  3,  shows  that 
3  of  these  parts  are  taken. 

Analyze  the  following: 


2. 

3. 
4. 


■2  . 

3  > 

5  . 

e  > 
3 . 

7  > 


4 

7" 

K      4  .       3 
°'    B">    77* 

8. 

9  .    7 

15?      18"' 

4 
5" 

6      7  .      8_ 
U*     9  >     11' 

9. 

13.  16 
2  1  )     2i' 

2 

7      12.      8 
1  '      13  J      14' 

10. 

21.34 

1  1" 

"3  1   '     4  4* 

PRINCIPLES  OF  FRACTIONS. 

lOO-LWe  will  now  solve  a  number  of  problems,  and 
derive  some  of  the  principles  of  fractions. 
1.  Multiply  the  numerator  of  |  by  2. 


06 


COMMON    FRACTIONS. 


Solution. — Multiplying  the  numerator  of  |         operation. 
by  2,  we  have  6  fifths,  which  is  2  times  as  great  |X2  ==  | 

as  3  fifths.     Hence  the  following 

Principle  I. — Multiplying  the  numerator  of  a  fraction 
by  any  number  multiplies  the  fraction  by  that  number. 


Multiply  the  fraction 

2.  |  by  5.  Ans.  L«. 

3.  |  by  7. 

4.  }|  by  8. 

5.  if  by  11. 


Multiply  the  fraction 

6.  \  §  by  14. 

7.  ||  by  18. 

8.  if  by  17. 

9.  Jf  by  20. 


OPERATION 
9  —  2 


4 
~5 


1.  Divide  the  numerator  of  J  by  2. 

Solution. — Dividing  the  numerator  of  f  by 
2,  we  have  2  fifths,  which  is  1  half  of  4  fifths. 
Hence  the  following 

Principle  II. — Dividing  the  numerator  of  a  fraction  by 
any  number  divides  the  fraction  by  that  number. 


Divide  the  fraction 
2.  f  by  3.  Ans.  f. 

3. 
4. 
5. 


|  by  4. 
J?  by  5. 

14  V  7. 


Divide  the  fraction 
6.   if  by  4. 
7-  if  by  9. 

8.  iff  by  12. 

9.  |Sf  by  32. 


OPERATION. 
3  V  1  —  1 


1.  Multiply  the  denominator  of  f  by  ! 

Solution. — Multiplying  the  denominator  by 
2,  we  have  3  eighths,  which  is  one-half  as  much 
as  3  fourths,  since  eighths  are  only  half  as  large 
as  fourths.     Hence  the  following 

Principle  III. — Multiplying  the  denominator  of  a  frac- 
tion by  any  number  divides  the  fraction  by  that  number. 


Divide  the  fraction 

Divide  the  fraction 

2. 

i  by  4. 

Ans.  A. 

1    - 

7. 

IfbyS 

3. 

H  b>'  1- 

Ans.  i|. 

8. 

1  by  6. 

4. 

!  by  5. 

9. 

11  by  12. 

5 

H  by  7. 

10. 

t9u  by  11. 

6. 

1  by  3. 

11. 

if  by  13. 

COMMON    FRACTIONS.  97 

1.  Divide  the  denominator  of  f  by  2. 

Solution. — Dividing  the  denominator  by  2,  we  have  3  halves,  and 
S  halves  is  twice  as  great  as  3  fourths,  since  halves  are  twice  as  large 
as  fourths.     Hence  the  following 

Principle  IV. — Dividing  the  denominator  of  a  fraction 
by  any  number  multiplies  the  fraction  by  ihat  number. 

Multiply,  by  dividing  the  denominator, 


2.  §  by  2. 

3.  13  by  6. 

4.  |  by  3. 

5.  i|  by  9. 

6.  |  by  4. 

7.  a?  by  7. 


8.  j%  by  5. 

9.  f|  bv  12. 

10.  j§  by  6. 

11.  <J  by  19. 

12.  ii  by  10. 

13.  TV?  by  36. 


1.  Multiply  both  numerator  and  denominator  of  §  by  2. 

Solution. — Multiplying  both  numerator  and  de-     operation. 
nominator  by  2,  we  have  £  ;  and  this  equals  f ,  since       f  X  f  ~  t 
we  both  multiplied  and  divided  §  by  2,  and  hence 
did  not  change  its  value.     Hence  we  have  the  following 

Principle  V. — Multiplying  both  numerator  and  denomi- 
nator of  a  fraction  by  the  same  number  does  not  change  the 
value  of  the  fraction. 

2.  Multiply  both  numerator  and  denominator  of  |  by 
3;  |  by  6;  |  by  5 ;  §  by  3 ;  i»  by  9 ;  j|  by  6. 

1.  Divide  both  numerator  and  denominator  of  |  by  2. 

Solution. — Dividing  both  numerator  and  denomi-      operation. 
nator  by  2,  we  have  f ;  and  this  equals  |,  since  we      |-^|=| 
both  divided  and  multiplied  |  by  2,  and  hence  did 
not  change  its  value.     From  this  we  have 

Principle  VI. — Dividing  both  numerator  and  denomina- 
tor by  the  samenumber  does  not  change  the  value  of  the  fraction. 

2.  Divide  both  numerator  and  denominator  of  g  by  2; 
l%by4;  J>3  by  3/  ig  by  2 ;  Jjby4j  IJbylO:  |$byl$ 

9 


98 


REDUCTION    OF    FRACTIONS. 


REDUCTION  OF  FRACTIONS. 

101.  Reduction  of  fractions  is  the  process  of  changing 
their  form  without  changing  their  value. 

Case  I. 

102.  To  reduce  mixed  numbers  to  fractions. 

1.  How  many  thirds  in  4#  ?  operation. 

Solution. — In  1  there  are  f,  and  in  4  there  4| 

are  4  times  f,  which  are  ^2,  and  ^  -f  §  equal  3 

J34.     From  this  solution  we  have  the  following  14  thirds  =  i-4. 

Rule. — Multiply  the  tohole  number  by  the  denominator, 
add  the  numerator,  and  write  the  denominator  under  the 
result. 


Reduce 

to 

improper 

fractions 

2. 

Ans. 

23 

5  • 

7.  181. 

3. 

1  4- 

8.  21}. 

4. 

Q5 

9.  19JA. 

5. 

7§. 

10.  25TV 

6. 

131. 

11.  35}§. 

Casi 

s  II. 

Ans.  9,-3. 


103.  To  reduce  improper  fractions  to  whole  or 
mixed  numbers. 

1.  How  many  ones  in  \5  ? 

Solution. — In  one  there  are  4  fourths,  and 
in  15  fourths  there  are  as  many  ones  as  4  is 
contained  times  in  15,  which  are  3-|.  From 
this  solution  we  have  the  following 


is 


OPERATION 

=  15 


4  =  3|. 


Rule. — Divide  the  numerator  by  the  denominator,  and 
the  quotient  will  be  the  whole  or  mixed  number. 

Reduce  to  whole  or  mixed  numbers 

7. 
8. 


2. 
3. 
4. 
5. 
6. 


9 

4"* 
1  1 

3  ' 
1  9 

5  * 
32 

4 

¥■ 


Ans.  2\ 


9. 
10. 
11. 


47 

e  • 

92 

1  !* 

2  5 

i  2" 

2  35 
77  • 

7  2_4 
3  J   ' 


Ans.  Y 


REDUCTION    OF    FRACTIONS.  99 

Case  III. 
10  1.  To  reduce  fractions  to  liiguer  terms. 

1.  How  many  twelfths  in  |? 

Solution. — Multiplying  both  numerator  and  operation. 

denominator  of  a  fraction  by  the  same  nuin-        3 3  X  3  __  _<?__ 

ber  does  not  change  its  value,  Prin.  V. ;  hence,  4  ^  3 

multiplying  both  numerator  and  denominator  by 

3,  and  we  have  f  =  T92.     From  this  solution  we  have  the  following 

Eule. — Multiply  both  numerator   and  denominator  b§ 
any  number  which  will  give  the  required  denominator. 
Ecduce  Reduce 


2.  f  to  12ths.     Ans.  T83. 

3.  §  to  30ths.     Ans.  §§. 

4.  I  to  16ths. 

5.  T90  to  20ths. 

6.  I  to  27tks. 


7.  ]±  to  36ths. 

8.  }|  to  GOths. 

9.  1  to  81sts. 

10.  }|  to  80ths. 

11.  if  to  8-ltks. 


Case  IV. 

105.  To  reduce  a  fraction  to  lower  terms. 

106.  A  fraction  is  reduced  to  lower  terms  when  it  is 
reduced  to  one  having  a  smaller  numerator  and  denomi- 
nator. 

lot.  A  fraction  is  in  its  lowest  terms  when  it  cannot 
be  reduced  to  any  lower  terms. 

1.  Reduce  T%  to  fourths. 

Solution. — Dividing  both  numerator  and  de-         operation. 
nominator  of  a  fraction  by  the   same  number  3)&  =  f 

does  not  change  its  value,  Prin.  YI. ;  hence,  to 

reduce  T9T  to  fourths  wc  divide  both  numerator  and  denominator  by 
3,  and  we  have  |.     From  this  solution  we  have  the  following 

Rule. — 1.  To  reduce  a  fraction  to  lower  terms,  divide 
both  numerator  and  denominator  by  the  same  number  or 
numbers. 

2.  To  reduce  to  lowest  terms,  divide  in  this  way  until 
the  fraction  cannot  be  reduced  any  lower. 


100 


COMMON    DENOMINATOR. 


Reduce  to  lowest  terms         Reduce  to  lowest  terms 


2. 

3. 
4. 
5. 
6. 


f>      1  4 

"S5     VI  1  * 


8 
i  2  J 

1  0 
I  2> 

1  6 
24? 

2Bj 


12 

1  8* 

2  5 
30 

18 
2  7' 

48 
84* 


Ans.  § 

7 

Ans.  f . 

8 

Ans.  |. 

9 

10 

11 

24 
40' 
70 
8  0' 

4  5 

5  0' 
9  9 

108' 

96 


27 


Ans.  ?. 

0 


84 

108 
I  Z  0~ " 

1  2  1 
732' 
1  44 


10  4'    156' 


108 
1 


Case  V. 
To  reduce  compound  fractious  to  simple. 


What  are  f  of  %  ? 


Solution. — \  of  f 


-X,  since  mul- 
tiplying  the  denominator  of  a  fraction 
by  3  divides  the  fraction  by  3 ;  and  if 
of  f  =  T%,  f  of  f  equals  2  times  T4, 


OPERATION. 

2V4 
3X0       3X5      T* 


15' 


which  are  -A-     From  this  solution  we  have  the  following 

RULE. — Multiply  the  numerators  together,  and  the  de- 
nominators together. 


2. 

3. 
4. 
5. 
6. 


What  is 
-2  of  2? 

5  0f   7? 

6  U1     9  ' 

4    0f   IS? 

7  Ui     15" 

£  Of  A-i  ? 
9  Ui    1  2  * 

*   Of  -9-  ? 

8  Ui     10* 


Ans. 


21 

32" 


7. 

8. 

9. 
10. 
11. 


What  is 

11  of  A6? 

12  Ui  33 • 


3   0f    5   0f   7? 
7  0f 


9     Of 
TO   U1 


li? 
1  6  " 


%  Of  2  Of    '  7  ? 


8 


7  8 


5   of  -3  Of   i  4-  ? 
3    Ui     7    ^       3  tS  ' 


12.  A  had  |  of  a  ton  of  hay,  and  sold  his  neighbor  J 
of  it ;  how  much  did  he  sell  ? 


f  of  a  ton  of  hay,  and 


OPERATION. 


fXf  =  T8sAns. 


Solution. — If  A  had  f 
sold  his  neighbor  §  of  it,  he  sold  his  neighbor 
I  of  I  of  a  ton,  which  is  T83  of  a  ton. 

13.  A  boy  picked  §  of  a  bushel  of  strawberries,  and 
sold  §  of  them  ;  how  many  did  he  sell  ?     Ans.  {§,  or  §. 

14.  A  man  had  §  of  a  bushel  of  barley,  and  sold  j  of 
it ;  how  much  did  he  sell  ?  Ans.  |. 

15.  A  little  girl  had  J  of  a  melon,  and  gave  her  brother 
4  of  it ;  how  much  did  her  brother  receive  ?      Ans.  £. 

16.  Says  Jennie  to  Kate,  My  father  owns  f  of  f  of  f 
of  a  ship  j  what  part  of  the  ship  did  he  own  ?     Ans. 


o 

5' 


COMMON    FRACTIONS.  101 

COMMON    DENOMINATOR. 
109.  Fractions  have  a   Common  Denominator  when 
they  have  the  same  number  for  a  denominator. 
1.  Reduce  |  and  i  to  a  common  denominator. 

Solution. — Multiplying  both  numerator  and         operation. 

denominator  of  f  by  5,  the  denominator  of  -f,        3X*>    _  15 

we  have  |4;  and  multiplying  both  numerator  "4X5  "^ 

and  denominator  of  f  by  4,  the  denominator  of  4       4X4       16 

f,  we  have  ^§;  and  this  makes  the  fractions  have  z       5X4       20 
the  same  denominator;  hence  the  following 

Rule. — Multiply  both  numerator  and  denominator  of 
each  fraction  by  the  denominators  of  the  other  fractions. 

Or,  Multiply  both  numerator  and  denominator  of  each 
fraction  by  any  number  that  will  make  the  denominators 
ilike. 

Reduce  to  a  common  denominator 


2. 

-§  and  |.     Ans. 

1  0 

1  5  5 

1  2 
1  5' 

7. 

12    ancl    13 
1  3    allu    14' 

3. 

|  and  {.     Ans. 

24 

30? 

2  5 

"3  0* 

8. 

|3  j,  and  |. 

4. 

I  and  j. 

9. 

3    5    anci  4 
4,  6,  aiiu  5. 

5. 

|  and  f . 

10. 

4    §    and  5 

6. 

9    arid  -8- 

11. 

1   -G    and  1 

65   7'  a    u   6* 

ADDITION   OF   FRACTIONS. 
110.  Addition  of  Fractions  is  the  process  of  finding 
the  sum  of  two  or  more  fractions. 

Case  I. 
To  add  when  the  denominators  are  alike. 

1.  What  is  the  sum  of  I  and  io  ? 

Solution. — 2  fifths  plus  4  fifths  equals  6  fifths,         operation. 
which  equals  li.  f  +  f  =  f  =  H 

2.  What  is  the  sum  of  §  and  §  ?  Ans.  §. 

3.  What  is  the  sum  of  §  and  J  ?  Ans.  §,  or  1J. 
4  What  is  the  sum  of  §  and  J  ?  Ans.  H- 
5.  What  is  the  sum  of  I  and  |  ?  Ans.  \5,  or  If. 

9* 


102 


COMMON    FRACTIONS. 


6.  Mary  had  §  of  a  dollar  and  Sarah  had  f  of  a  dollar; 
how  much  did  they  both  have  ? 

7.  Lucy  gave  me  §  of  a  peach,  and  Fanny  gave  me  | 
of  a  peach;  how  much  did  I  receive?  Ans.  1^. 

8.  George  and  Susie  had  each  J  of  a  pine-apple ;  how 
much  had  they  together?  Ans.  If. 

9.  If  I  walk  |  of  a  mile  and  ride  |  of  a  mile,  how  far 


do  I  go  ? 


Ans.  11  mile. 


10.  A  had  |  of  a  dollar,  B  had  J  of  a  dollar,  and  C 
had  |  of  a  dollar ;  how  much  had  they  all  ? 


Case  II. 
To  add  when  tlae  denominators  are  unlike. 

1.  What  is  the  sum  of  §  and  |  ? 
Solution. — We  first  reduce  the  fractions  to  a 


common  denominator:  -| 


8    •    3 
12  '  ? 


T9j ;  8  twelfths 


plus  9  twelfths  are  17  twelfths.    Hence  f  +  f  - 


12' 


OPERATION. 

2  _|_  3  

3  T  4 

T2  +  T%  =  TT 


From  this  we  have  the  following 

Eule. — Reduce  the  fractions  to  a  common  denominator; 

add  the  numerators,  and  place  the  sum  over  the  common 

denominator. 

N0TE. — Reduce  each  fraction  to  its  lowest  terms  before  reducing 
to  a  common  denominator,  and  also  the  result  after  addition. 


Find  the  sum  of 


2. 

3. 
4. 

5. 
6. 

7. 
8. 


and 


2 


|  and  \. 
|  and  |. 
|  and  § . 
|  and  g. 
%  and  Tq 
i  and  §. 


Ans. 
Ans. 


16 
1  5* 

31 

20' 


Find  the  sum  of 
Ans. 

Ans. 


|  and  f . 
9  anc*  TO* 

4   ar]Cl    12 
6   <UJU    14' 


97 
5  6' 

31 

3  0* 


9. 
10. 

11.  o 

12.  f|  and  Jf. 

13.  if  and  flf. 

14.  1«  and  -}f . 

15.  i§  and  U. 

of  a  pie;  how  much 


16.  A  has  f  of  a  pie,  and  B  f 
have  they  both  ? 

17.  B  having  j  of  a  ton  of  hay  bought  %  of  a  ton; 
Vow  much  had  he  then? 


COMMON    FRACTIONS.  103 

18    Henry  owned  §  of  a  vessel,  and  bought  \  of  the 
ressel ;  how  much  did  he  then  own  ? 

19.  A  had  7f  dollars,  and  B  gave  him  8|  dollars  ;  how 
many  had  he  then  ? 

Note.— Add  the  7  and  8,  and  then  add  £  and  f;  8  +  7  =  15: 

!  +  f=A  +  A  =  H  =  1A'  15+1T5.  =  10TV  Ans. 

20.  A  had  25 r  acres  of  land,  and  then  bought  17| 
acres ;  how  many  had  he  then  ? 

21.  B  had  57|  dollars,  and  C  had  96|  dollars;  what 
was  the  sum  of  their  money  ? 

22.  C  sold  9G|  yards  of  cloth,  and  then  had  147T99 
yards  left ;  how  much  had  he  at  first  ? 

SUBTRACTION  OF  FRACTIONS. 

111.  Subtraction  of  Fractions  is  the  process  of  find- 
ing  the  difference  between  two  fractions. 

Case  I. 
To  subtract  when  the  denominators  are  alike. 

1.  Subtract  |  from  §. 

Solution.— 3  eighths  subtracted  from  7  eighths         operation. 
leave    4    eighths,    and  f   reduced    to   its   lowest      f  — 1  =  |,  or  £ 
terms  equals  \. 

2.  Subtract  f  from  f . 

3.  Subtract  \  from  f. 

4.  Subtract  |  from  f . 

5.  Subtract  T53  from  }£. 

6.  Mary  had  g  of  an  apple,  and  gave  away  |  of  an 
apple  ;  what  part  of  an  apple  had  she  left  ?       Ans.  £. 

7.  Peter  found  i  of  a  dollar,  and  spent  §  of  a  dollar ; 
what  part,  of  a  dollar  had  he  left? 

8.  If  I  buy  \\  of  a  ton  of  hay  and  sell  j5.2  of  a  ton, 
what  part  of  a  ton  will  I  have  left  ? 

i).  Peter  has  T%  of  a  dollar,  John  /„  of  a  dollar,  and 


104  COMMON    FRACTIONS. 

Jacob  T4o  of  a  dollar ;  how  much  more  have  Peter  and 
John  than  Jacob  ?  Ans.  $^. 

10.  Mary  has  §  of  a  dollar,  Sarah  i  of  a  dollar,  and 
Jane  §  of  a  dollar ;  how  much  more  have  Mary  and 
Sarah  than  Jane  ?  Ans.  $1. 

Case  II. 
To  subtract  when  the  denominators  are  unlike. 

1.  Subtract  |  from  f. 

Solution. — We  will  first  reduce  the  fractions  operation. 

to  a  common  denominator.    f  =  xf  and  f =1&  t —  I 

and  12  fifteenths  minus  10  fifteenths  is  2  fifteenths.  if  —  \% 
From  this  solution  we  have  the  following 


2 
T5 


Eule. — Reduce  the  fractions  to  a  common  denominator, 
subtract  the  numerators,  and  place  the  result  over  the  com- 
mon  denominator. 

Note. — Reduce  each  fraction,  and  also  the  difference,  to  its  lowest 
terms. 


Subtract 

2.  I  from  f .       Ans.  T^. 

3.  |  from  |.       Ans.  y1^. 

4.  |  from  {. 

5.  |  from  §. 

6.  |  from  §. 


Subtract 

8.  §  from  ig.    Ans.  5\. 

9.  £  from  J.       Ans.  Jg. 

10.  f  from  jf. 

11.  |  from  {I. 

12.  £  from  TV 

13.  II  from  U. 


"r.  «  from  {I. 

14.  Mary  had  |  of  a  dollar  and  gave  away  \  of  a  dol- 
lar ;  how  much  had  she  left  ? 

Solution. — If  Mary  had  f  of  a  dollar,  and 
gave  away  \  of  a  dollar,  she  had  left  the  differ-         operation. 

ence  between  |  of  a  dollar  and  £  of  a  dollar,  f  —  \  — 

which  by  reducing  to  a  common   denominator  \%  —  ^  =  ^ff  Ans. 
and  subtracting,  we  find  to  be  -fa  of  a  dollar 

15.  Willie  gave  Sallie  §  of  a  quart  of  peanuts,  and 
Sallie  gave  him  back  |  of  a  quart ;  what  part  of  a  quart 
did  Sallie  keep  ? 


COMMON    FRACTIONS.  105 

16  A  has  2  of  a  pie;  if  he  gives  B  |  of  a  pie,  how 
much  vi  ill  remain  ? 

17.  B  bought  T90  of  a  ton  of  hay,  and  sold  C  f  of  a 
ton;  how  much  did  B  retain? 

18.  C  owned  ]  of  a  vessel,  H  bought  J  of  this,  and 
then  sold  I  of  what  he  bought:  how  much  did  H  keep? 

19.  The  sum  of  two  fractions  is  £{j,  and  one  fraction 
is  li  •  what  is  the  other  fraction  ? 

20.  If  D  had  |  of  a  certain  sum  of  money,  and  then 
earned  f  of  the  same  sum,  arid  then  spent  J  of  the  sum, 
how  much  remained  ? 

21.  The  sum  of  three  fractions  is  If,  and  the  other 
two  fractions  are  -\  and  \  ;  what  is  the  third  fraction  ? 

22.  A  has  |  of  a  sum  of  money;  he  owes  B  §  of  that 
sum  and  C  1  of  the  sum:  how  much  will  remain  after 
paying  his  debts  ?  Ans.  /<j. 

PRACTICAL  PROBLEMS 
in  Addition  and  Subtraction  of  Fractions. 
1.  Subtract  4f  from  74. 

4  "*  OPERATION. 

Solution.— 1\   equals    6 -f  £ -f  |  =  6£;    4|  7£  =  6f 


Dtr 

acted  from  6|  leaves  2|. 

43 

2f  Ans. 

Subtract 

Subtract 

2. 

5|  from  9J. 

5. 

I65 from  2(H 

3. 

7|  from  10$. 

6. 

19f  from  30§. 

4. 

8|  from  13f. 

7. 

24|  from  36|. 

8.  A  had  17?  dollars,  and  gave  B  12| ;  how  much 
remained  ? 

9.  A  has  6|  dollars,  and  B  has  7^  dollars;  how  much 
have  both  ? 

10.  A  read  4^  pages  one  day  and  7  J  another  day ; 
how  many  pages  did  he  read  in  the  two  days? 

11.  Mary  had  20  dollars;  she  gave  her  brother  $9| 
and  her  sister  $l'l  ;  how  much  remained  ? 


106  COMMON    FRACTIONS. 

12.  William  having  $100  gave  $26-  to  the  poor  and 
spent  S18|  for  clothing;  how  much  remained? 

13.  My  father  gave  me  $3|,  my  mother  gave  me  $5|, 
and  I  then  gave  my  sister  $4| ;  how  much  remained? 

14.  What  is  the  sum  of  h  of  -J  and  |  of  \  ;  and  what, 
also,  is  their  difference  ? 

15.  ]  had  $24,  and  gave  \  of  it  to  my  sister  and  \  of 
it  to  my  brother ;  how  much  remained  ? 

16.  Sarah  had  $23,  and  gave  \  of  it  to  the  poor  and  § 
of  it  for  a  dress ;  how  much  remained  ? 

17.  Henry's  father  gave  him  $16|,  and  his  mother 
gave  him  $18£j  he  then  spent  f  of  it;  how  much  re- 
mained? 

18.  A  man  had  $24f ,  and  then  earned  $161,  and  then 
spent  A  of  it;  how  much  remained? 

19.  Peter  had  $17-J,  and  then  lost  $11  -J,  and  then 
earned  $14  ;  how  much  had  he  then  ? 

20.  Harold  had  $|,  then  lost  $|,  and  then  earned  $|  ; 
how  much  had  he  then  ? 

MULTIPLICATION  OF  FRACTIONS. 

112.  Multiplication  of  Fractions  is  the  process  oi 
multiplying  when  one  or  both  terms  are  fractions. 

Case  I. 

113.  To  multiply  a  fraction  by  a  whole  number. 

1.  Multiply  |  by  4. 

Solution.— 4  times  §  equal    \°,  according  to         operation. 


Prin.    I.     Or,  4   times   f   equal     f,    since   di-         f  X  4  =  \° 
viding    the   denominator    multiplies    the   frac-     or,  |  X  4  —  I 
tion,   according  4«  Prin.   IV.      From  this  we 
have  the  following 

Rule—  To  multiply  a  fraction  by  an  integer,  multiply 
the  numerator  or  divide  the  denominator  by  the  integer-. 


COMMON    FRACTIONS. 


107 


Multiply 

Multiply 

2. 

A  by  5. 

Ans.  J. 

6. 

13  by  9. 

Ans.  8J 

3. 

Uby4. 

Ans.  y. 

7. 

i?  by  7. 

Ans.  6? 

4. 

I^by7. 

8. 

P  by  12. 

5. 

i!  by  3. 

9. 

mby36. 

10.  A  has  }J  of  a  ton  of  hay,  and  B  has  3  times  as 
nuch;  how  much  have  both? 

Case  II. 
114.  To  multiply  a  whole  number  by  a  fraction. 

1.  Multiply  8  by  f ;  also  by  4§. 


Solution  1. — 8  multiplied  by  ^  equals  %  of  8, 
or  f,  and  8  multiplied  by  §  equals  2  times  f, 

16 


or  * 


Solution  2. — We  multiply  8  by  2  and  divide 
by  8,  and  have  51;  then  multiply  by  4  and  add 
the  product  32  to  51,  making  371.  Hence,  in  a 
mixed  number  we  multiply  first  by  the  fraction, 
and  then  by  the  integer.  From  these  solutions 
we  have  the  following 


OPERATION. 

8X2 


OPERATION. 
8 

^3 


3)16 


32 
3Ti 


Rule. — Multiply  the  whole  number  by  the  numerator  of 
the  fraction,  and  divide  the  product  by  the  denominator. 


Multiply 

2.  16  by  |. 

3.  18  by  §. 

4.  12  by  l. 

5.  20  by  2TV 

6.  35  by  45. 


Ans.  12. 
Ans.  15. 


Multiply 

7.  45  by  §. 

8.  43  by  g. 

9.  28  by  5}. 

10.  76  by  4|. 

11.  85  by  8TV 


Ans.  40. 
Ans.  35| 


12.  A  has  18  tons  of  hay,  and  B  has  4|  times  as  much, 
plus  31  tons;  how  much  has  B? 

Case  III. 
115.  To  multiply  a  fraction  by  a  fraction. 

1.  Multiply  I  by  j. 


108 


DIVISION    OF    FRACTIONS. 


|  multiplied  by  \  equals  \  of 

III. ;   and 
|  multiplied  by  f  equals  3  times  -fa  or  fa. 


Solution. — 

which  is  fa,  according  to  Prin. 


OPERATION. 


3  V   3  

"5  A  £  — 


3  X  3  _  9 

—  tv 

5X4 


Eule. — Multiply  the  numerators  together  for  the  numera- 
tor, and  the  denominators  together  for  the  denominator  of 
the  product. 

Note. — Reduce  the  result  to  its  lowest  terms. 
What  is  the  product  of 


2. 
3. 
4. 
5. 
6. 


I  by  |? 

I  by 

ft? 

6  * 

J>    ? 

1  4  * 

15? 

1  6 


4? 

t9o  by° 
iiby 

M*y 


Ans. 
Ans. 


3 

TO* 

7 
TO"' 


7. 

8. 

9. 
10. 
11. 


I?  by 

by 


2  5 
2  9 


V 

28  ? 
25  ' 
17  ? 
3  5  ' 


Ans 

Ans. 

Ans.  : 


is 

24* 

1  6 
TO- 
SS 
103* 


1  by  |  of  |? 


I  of 


5    U J    2  8' 


12-  A  has  |  of  a  ton  of  hay,  and  B  has  f  as  much 
plus  2|  tons ;  how  much  has  B  ? 

DIVISION  OF  FRACTIONS. 

116.  Division  of  Fractions  is  the  process  of  dividing 
when  one  or  both  terms  are  fractional. 


OPERATION. 


8  ^_4 

9  •     * 


Case  I. 
117.  To  divide  when  the  dividend  is  a  fraction, 

1.  Divide  |  by  4. 

Solution. — |  divided  by  4  equals  f ,  according  to 
Prin.  I.  When  the  numerator  will  not  contain- 
Ihe  divisor,  we  multiply  the  denominator,  accord- 
ing to  Prin.  III. 

Rule. — Divide  the  numerator,  or  multiply  the  denomt 
nator,  by  the  divisor. 

Divide 

t9o  by  3.  Ans. 

by  4.  Ans. 

by  6. 


2. 
3. 
4. 

5.  _9T  by  4. 

6.  |  by  a 


_8_ 
1  1 
12 
1  3 


3 

TO* 
_2_ 

i  r 


7. 

8. 

9. 
10. 
11. 


Divide 

H  by  7. 
if  by  5 

V8  by  8. 

31  by  9. 
5|  by  12. 


Ans. 

Ans. 


12 
91' 
i  6 

5  5' 


MISCELLANEOUS    EXAMPLES.  ■  109 

12.  A  gave  3A  dollars  to  6  little  girls ;  how  much  did 
each  receive  ? 

Case  II. 
118.  To  divide  when  the  divisor  is  a  fraction. 

1.  Divide  f  by  -1. 

Solution. — |  divided  by  1  equals  f,  hence  -J         operation. 
divided  by  A  equals  5  times  f ,  and  -|  divided  by         |  -r-  f  = 
£  equal     £  ot  5  times   f,   or  J  times  f,  which         |Xf  =  xf 
equal     ^|.     Hence,  we  see  the  divisor  becomes 
inverted,  and  we  have  the  following 

Rule  — Invert  the  divisor,  and  multiply  the  dividend  by 
the  resulting  fraction. 

Divide  Divide 


2. 

ffcyf 

Ans.  |. 

8. 

14   bv  -9- 

Ans.  1J. 

o 
O. 

i  by  I 

Ans.  f . 

9. 

2.1    hV   14 

3  2    UJ     lS' 

Ans.  f  J. 

4 

I  by  §. 

Ans.  |i. 

10. 

2  0   |)V   18 
2  1    UJ     3  5* 

Ans.  Iff. 

5. 

t9o  *y  !• 

11. 

16    V)V     8 

Ans.  2T8T. 

6. 

H  by  {. 

12. 

15    u  J     3  5* 

7. 

I?  by  fj. 

13. 

3  2   Kv   4  8 
3  6    UJ     5  0' 

14.  How  many  yards  of  cloth  at  |  of  a  dollar  a  yard 
can  you  buy  for  4^  dollars  ? 

MISCELLANEOUS   EXAMPLES. 

1.  Reduce  32|  to  an  improper  fraction. 

2.  Reduce  47TfiT  to  an  improper  fraction. 

3.  Reduce  4T°48  to  a  mixed  number. 

4.  Reduce  ^f8  to  a  mixed  number. 

5.  Reduce  |J|  to  its  lowest  terms. 

6.  Reduce  y4^  to  its  lowest  terms. 

7.  Reduce  f  of  -f  of  §  to  a  simple  fraction. 

8.  Reduce  |  of  ||  of  -^8  to  a  simple  fraction. 

9.  Reduce  §,  |,  and  J  to  a  common  denominator. 

10.  Reduce  |,  |,  and  -j  Q  to  a  common  denominator 

11.  Find  the  sum  of  J,  J,  i,  and  J. 
12-   Find  the  sum  of  f,  j,  <,  and  {. 


110  •  ANALYSIS. 

13.  Subtract  §  of  |  from  §  of  |. 

14.  Subtract  ?  of  f.  from  the  sum  of  ?  and  |. 

4  0  o  4 

15.  Multiply  the  sum  of  ^  and  ]  by  |  plus  i. 
1G.  Multiply  I  -J-  |  by  the  sum  of  §  and  |. 

17.  Divide  I -{- 1  by  the  sum  of  \  and  f . 

18.  Divide  the  sum  of  §  and  J-  by  |  minus  §. 

19.  What  cost  24  apples  at  |  of  a  cent  each? 

20.  What  cost  45  oranges  at  2§  cents  apiece  ? 

21.  How  much  cost  16  j  yards  of  cloth  at  $6  a  yard  ? 

22.  What  cost  16-  yards  of  muslin  at  12  J  cents  a  yard? 

23.  If  one  yard  of  cloth  cost  $9,  how  many  yards  can 
you  buy  for  $48  ?  Ans.  5  J  yds. 

24.  How  many  yards  of  muslin  at  16J  cents  a  yard 
can  you  buy  for  208^  cents?  Ans  12^  yds. 


AEITHMETICAL   ANALYSIS. 

119.  Analysis  is  the  process  of  solving  problems  by 
a  comparison  of  their  elements.  In  comparing,  we 
reason  to  the  unit  and  from  the  unit,  the  unit  being  the 
basis  of  the  reasoning  process. 

Case  I. 

120.  To  pass  irons,  one  integer  to  another. 

1.  If  5  cows  cost  $80,  what  will  7  cows  cost  at  the 
same  rate  ? 

OPERATION. 

Solution. — If  5  cows  cost  $80,  one  cow  costs  |        5)80 
of  $80,  which  is  $16,  and  7  cows  will  cost  7  times  16 

$16,  which  are  $112.  _7 

112  Ans. 

2.  If  6  hens  cost  186  cents,  what  will  9  hens  cost  at 
the  same  rate? 

3.  If  5  pigs  cost  $35,  what  will  11  pigs  cost  at  the 
same  rate  ? 


ANALYSIS. 


Ill 


4.  If  8  horses  cost  $1200,  what  will  12  horses  cost  at 
the  same  rate? 

5.  If  7  yards  of  cloth  cost  842,  what  will  25  yards 
cost  at  the  same  rate  ? 

6.  How  much  must  I  pay  for  3G  cows,  at  the  rate  of 
7  cows  for  196  dollars  ? 

7.  What  will  17  books  cost,  at  the  rate  of  8  books  for- 
810.80? 

8.  A  man  bought  72  ducks  at  the  rate  of  $21  for  7; 

what  did  they  cost? 

9.  If  a  man  can  walk  324  miles  in  9  days,  how  far 
can  he  walk  in  69  days? 

10.  In  26  years  there  are  9490  days ;  how  many  days 
are  there  in  75  years  ? 

11.  In  5  square  miles  there  are  3200  acres  ;  how  many 
acres  in  64  square  miles  ? 

12.  If  a  car  run  2736  miles  in  18  days,  how  far  will 
it  run  in  54  days  ? 

Case  II. 
121.  To  pass  from  a  fraction  to  an  integer. 

1.  If  |  of  an  acre  of  land  cost  $96,  what  will  one  acre 
cost? 

OPERATION. 

Solution. — If  §  of  an  acre  cost  $96,  I  of  an  §  —  $96 

acre  cost  A  of  $96,  or  $48,  and  if  A  of  an  acre  cost  i  ==  $48 

$48,  |  of  an  acre,  or  one  acre,  will  cost  3  times  f  =  $144  Ans. 
$48,  or  $144. 

2.  If  |  of  a  sum  of  money  is  $72,  required  the  sum. 

3.  If  |  of  the  cost  of  a  cow  is  $25,  required  the  cost 
of  the  cow. 

4.  What  cost  2  boxes  of  raisins,  if  -?  of  a  box  cost  6 
dollars  ? 

5.  What  is  the  distance  from  Lancaster  to  Philadel- 
phia, if  |  of  the  distance  is  51  miles? 

6    If  the  cost  of  |  of  an  acre  of  land  is  $120,  what 
will  4  acres  cost  at  the  same  rate? 


112  COMMON    FRACTIONS. 

7.  If  f  of  a  farm  cost  $7200,  what  will  the  whole 
farm  cost  at  that  rate  ? 

8.  How  much  will  7  loads  of  hay  weigh,  if  J  of  a 
load  weighs  840  pounds  ? 

9.  What  will  17  horses  cost  me,  if  f  of  the  price  of 
a  horse  is  93  dollars  ? 

10.  A  merchant  bought  236  barrels  of  flour  at  the 
rate  of  $8  for  §  of  a  barrel ;  how  much  did  they  cost 
him? 

Case  III. 
122.  To  pass  from  a  nnit  or  fraction  to  a  fraction, 

1.  If  one  barrel  of  flour  coHs  $12,  what  will  j  of  a 
barrel  cost  ? 

Solution. — If  one  barrel  of  flour  costs  $12,         operation. 
1  fourth  of  a  barrel  will  cost  \  of  $12,  or  $3,  4)12 

and  |  of  a  barrel  will  cost  3  times  $3,  or  $9.  3 

3 

9  Ans. 

2.  If  one  acre  of  land  is  worth  $125,  what  is  4  of  an 
acre  worth  ? 

3.  A  paid  $1650  for  a  pleasure-boat ;  how  much  would 
he  have  paid  if  he  had  given  §  as  much  ? 

4.  If  |  of  a  barrel  of  flour  cost  $8,  what  will  f  of  a 
barrel  cost  ? 

5.  If  there  are  40  pounds  in  |  of  a  bushel  of  clover- 
seed,  how  many  pounds  are  there  in  §  of  a  bushel  ? 

6.  If  there  are  50  pounds  in  |  of  a  bushel  of  wheat, 
how  many  pounds  are  there  in  j]  of  a  bushel  ? 

7.  If  there  are  49  pounds  in  J  of  a  bushel  of  rye, 
how  many  pounds  are  there  in  |  of  a  bushel? 

8.  If  there  are  147  pounds  in  |  of  a  barrel  of  flour, 
how  many  pounds  are  there  in  |  of  a  barrel  ? 

9.  If  there  are  154  cubic  inches  in  f  of  a  gallon,  how 
many  cubic  inches  in  ^  of  a  gallon  ? 


COMMON    FRACTIONS.  113 

10.  /f  there  are  1536  cubic  inches  in  §  of  a  cubic  foot, 
how  many  cubic  inches  in  ji  of  a  cubic  foot? 

Case  IT. 
123.   Given  a  fractional  part  and  the  remainder, 
to  find  the  whole. 

1.  A  man  spent  |  of  his  money,  and  then  had  $24  re* 
maining ;  how  much  money  had  he  at  first  ? 

Solution. — If  he  spent  §  of  his  money,  operation. 

there  remained  §  of  his  money  minus  f  of  f  —  f  =  §  ==  $24 
his  money,  which  is  §  of  his  money,  which  ^  =  $12 

is  S24.     If  |  of  his  money  is  §24,  I  of  his  $  =  $60  Ans. 

money  is  |  of  $24,  which  is  $12,  and  §  of 
his  money  is  5  times  $12,  or  $00. 

2.  A  man  spent  f  of  his  money,  and  then  had  $30  re- 
maining; how  much  had  he  at  first? 

3.  William  sold  I  of  his  hens,  and  then  had  60  remain- 
ing;  how  many  had  he  at  first  ? 

4.  Henry  sold  §  of  his  bank-stock,  and  the  remainder 
was  worth  $550  j  how  much  had  he  at  first  ? 

5.  After  giving  \  of  his  income  to  the  poor,  Samuel 
had  $960  remaining;  what  was  his  income  ? 

6.  A  pole  stands  J  in  the  mud  and  |  in  the  water,  and 
12  feet  in  the  air;  required  the  length  of  the  pole. 

7.  One-fourth  of  a  drove  of  animals  are  cows,  I  are 
pigs,  and  the  remainder,  132,  are  sheep;  how  many 
animals  in  the  drove? 

8.  Two-fifths  of  my  money  is  in  bank,  |  in  govern- 
ment bonds,  and  $480  in  cash;  what  was  my  money  ? 

9.  A  sold  J  of  his  land  to  B,  and  |  to  C,  and  then  had 
90  acres  remaining;  how  much  had  he  at  first? 

10.  A  man  walked  §  of  the  distance  from  Lancaster  to 
Philadelphia  one  day,  |  of  the  distance  the  next  day, 
and  the  remaining  distance,  22  miles,  the  third  day; 
how  far  did  he  walk  each  day  ?  Ans.  20  ;  28 ;  22. 

10* 


114  DECIMAL    FRACTIONS. 

SECTION    VI. 

DECIMAL   FRACTIONS. 

124.  A  Decimal  Fraction  is  a  number  of  the  deci- 
mal divisions  of  a  unit;  that  is,  a  number  of  tenths,  hun- 
dredths, etc. 

125.  A  decimal  fraction  is  usually  expressed  by 
placing  a  point  before  the  numerator  and  omitting  the 
denominator.     Thus,  .5   represents   T5D ;    .05   represents 


TOO'  e^c- 


126.  The  point  is  called  the  decimal  point,  or  separa- 
trix.  The  decimal  fraction  thus  expressed  is  called  a 
decimal. 

127.  This  method  of  expressing  decimal  fractions  is 
but  an  extension  of  the  method  of  notation  for  integers. 
This  method,  as  applied  to  integers  and  fractions,  is  ex- 
hibited in  the  following 


o 


NOTATION  AND  NUMERATION  TABLE. 


DO 

a 
a 

CO 

m  o 

o  *t 


CO       ,-j         CO 

Xl     2  o>  o  5  © 
6666666  6.  6666666 


05 

M 

^ 

C3 
CO 

3 

• 

-3 

^q 

02 

-2 

03 
0) 

CO 

co 

a 

CO 

3 
O 

O 

I 

S- 

H3 

CO 

o 

o 
3 

0) 

a 

5 
— | 

1 

g 

.— 

0) 

Eh 

i-M 

l-i-C 

H 

H 

<s 

H 

EXAMPLES    IN    NUMERATION.      ■ 

1.  Read  the  decimal  .36. 

Solution. — This  expresses  3  tenths  and  6  hundredths,  or,  since 
3  tenths  equals  30  hundredths,  and  30  hundredths  plus  6  hundredths 
equal  36  hundredths,  it  may  also  be  read  36  hundredths. 


DECIMAL    FRACTIONS. 


115 


128.  Hence  there  are  two  methods  of  reading  deci- 
mals, which  are  expressed  by  the  following  rules: — 

Eule  1. —  Commencing  at   tenths,  read   each  figure    in 
order  toward  the  right,  giving  it  Us  proper  denomination. 

Rule  2. — Read  the   decimal   as  a  whole   number,  and 
give  it  the  denomination  of  the  last  figure  on  the  right; 
numerating  toward  the  point  to  determine  the  numerator, 
and  from  the  point  to  determine  the  denominator. 
Read  the  following  decimals: — 


2. 

.45. 

6.  .046. 

10.  2.0123. 

3. 

.83. 

7.  .007. 

11.  4.2057. 

4. 

.126. 

8.  .3216. 

12.  13.0205. 

5. 

.324. 

9.  .1357. 

13.  27.0027. 

examples  in  notation. 
1.  Express  25  hundredths  in  the  form  of  a  decimal. 

Solution.—  25  hundredths  equals  2  tenths  and  5  hundredths,  and 
this  is  expressed  by  writing  a  decimal  point  before  25,  thus,  .25 
Hence  the  following  * 

Rule. —  Write  the  decimal  as  we  would  a  whole  number, 
placing  the  decimal  point  so  as  to  give  each  figure  its  proper 
place,  using  ciphers  after  the  decimal  point  if  necessary. 

Express  the  following  in  the  decimal  form. 

2.  Thirty-four  hundredths. 

3.  Seventy-five  hundredths. 

4.  Two-tenths  and  six-hundredths. 

5.  Twenty-five  thousandths. 

6.  Four-tenths  and  7-thousandths. 

7.  Seven-tenths  and  8-thousandths. 

8.  Five  hundred  and  25-thousandths. 

9.  Three-tenths  and  7  ten-thousandths. 

10.  Four  hundredths  and  96  millionths.    Ans.  .0400096. 

PRINCIPLES  OF  DECIMAL  NOTATION. 
129.  We   now  present   the  following  principles  of 
Decimals,  which  the  pupils  will  illustrate. 


116  DECIMAL    FRACTIONS. 

1.  Changing  the  decimal  point  one  place  toward  the  right 
multiplies  by  10;  two  places,  by  100,  etc. 

2.  Changing  the  decimal  point  one  place  toward  the  left 
divides  by  10 ;  two  places,  by  100,  etc. 

3.  Placing  a  cipher  between  the  decimal  point  and  a  deci- 
mal divides  the  decimal  by  ten. 

REDUCTION  OF  DECIMALS. 

130.  The  Reduction  of  Decimals  is  the  process  of 
changing  their  form  without  changing  their  value. 

There  are  two  cases  : — 

1.  To  reduce  decimals  to  common  fractions. 

2.  To  reduce  common  fractions  to  decimals. 

Case  I. 

131.  To  reduce  a  decimal  to  a  common  fraction. 

1.  Eeduce  .45  to  a  common  fraction. 

* 

Solution. — .45  expressed  in  the  form  of  a  operation. 

common  fraction,  is  T405o>  which,  reduced  to  its  .45  =  T\55 

lowest  terms,  equals  ^V     Hence  we  have  the  =  -fa  Ans. 
following 

RULE. —  Write  the  denominator  under  the  decimal,  omiU 
ting  the  decimal  point,  and  reduce  the  common  fraction  to 
its  lowest  terms. 

Eeduce  the  following  decimals  to  common  fractions. 


2. 

.35. 

Ans.  -Jq. 

6. 

9.75. 

Ans. 

»4« 

3. 

.48. 

Ans.  i$. 

7. 

.725. 

Ans. 

29 
4  0' 

4. 

.125. 

Ans.  |. 

8. 

.075. 

Ans. 

3 

40' 

0. 

.625. 

Ans.  g. 

9. 

.0125. 

Ans. 

1 
HO* 

Case  II. 
132.  To  reduce  a  common  fraction  to  a  decimal. 

1.  Eeduce  f  to  a  decimal. 


DECIMAL    FRACTIONS.  117 

Solution.— |  equals  \  of  3.     3  equals  30  operation. 

tenths,  and  \  of  30  tenths  is  7  tenths  and  2  f  =  \  of  3  = 

tenths  remaining.     2  tenths  equals  20  hun-  4)3.00 
dredths,   and  \  of  20  hundredths  is  5  hun-  .75 

dredths  ;  hence  f  =  .75.     From  this  we  have 
the  following 

Eule. — 1.  Annex  ciphers  to  the  numerator  and  divide  by 
the  denominator. 

2.  Point  off  as  many  places  in  the  quotient  as  there  are 
ciphers  annexed. 

Reduce  the  following  common  fractions  to  decimals. 


2.  4-  Ans.  .25 

4 


7.  TV  Ans.  .4375. 

1    D 


3.  |.  Ans.  .125. 

4    5 


5* 


5. 

6.  -V  Ans.  .3125. 


8.  T9S.  Ans.  .5625. 

1  o 
Q       1  1 

10    ±4 

11       P 
±x'     64* 


OPERATION 

7.5 

18.25 

21.36 

47.45 

ADDITION    OF   DECIMALS. 

133.  Addition  of  Decimals  is  the  process  of  finding 
the  sum  of  two  or  more  decimals. 

1.  What  is  the  sum  of  7.5,  18.25,  21.36  and  47.45? 

Solution. — "We   write    the  numbers    so    that 
figures   of  the   same  order   shall   stand  in  the 
same  column,  and  commence   at  the  right  to 
add.     5  hundredths,  plus  6  hundredths,  plus  5 
hundredths,  equal  16  hundredths,  which  equal 
1  tenth  and  6  hundredths;   we  write  the  6  hun-  94.56 

dredths,  and  add  the  1  tenth  to  the  next  sum. 

4  tenths,  plus  3  tenths,  plus  2  tenths,  plus  5  tenths,  are  14  tenths, 
and  the  1  tenth  added  are  15  tenths,  which  equals  1  unit  and  5 
tenths;  we  write  the  5  tenths,  and  add  the  1  unit  to  the  sum  of 
the  units,  etc. 

Rule. — 1.    Write  the  numbers  so  that  units  of  the  same 
order  shall  stand  in  the  same  column. 

2.  Add,  as  in  whole  numbers,  placing  the  decimal  point 
in  its  proper  place  in  the  sum. 


118  DECIMAL   FRACTIONS. 

2.  Find  the  sum  of  12.05,  33.24,  47.62,  96.47. 

3.  Find  the  sum  of  76.24,  89.45,  36.40,  85.75. 

4.  Find  the  sum  of  79.76,  85.08,  95.42,  237.675. 

5.  Add  18.79,  147.072,  856.709,  185.8761,  397.05784. 

6.  Add  59.874,  435.095,  672.328,  976.309,  8467.500843. 

7.  Add  together  9  and  7  tenths,  41  and  8  hundredths, 
75  and  54  hundredths,  128  and  187  thousandths. 

Ans.  254.507. 

8.  Add  together  76  and  49  hundredths,  127  and  49 
thousandths,  496  and  167  thousandths,  985  and  98  ten- 
thousandths,  and  99  and  99  hundred-thousandths. 

SUBTRACTION   OF   DECIMALS. 
134.  Subtraction  of  Decimals  is  the  process  of  find- 
ins:  the  difference  between  two  decimals. 

1.  From  67.35  take  42.63. 

Solution. — We  write   the   numbers   so   yiat         operation 
figures  of  the  same   order   stand  in  the  same  67.85 

column,  and  begin  at  the  right  to  subtract.     3  42.63 

hundredths  from  5  hundredths  leave     2  hun-  24.72 

dredths  ;    6  tenths  we  cannot  subtract  from  3 
tenths ;  we  therefore  take  1  unit  from  the  7  units,  which  with  3  tenths 
equal    13  tenths ;   then  6  tenths  from  13  tenths  leave    7  tenths,  etc. 

Rule. — 1.  Write  the  smaller  number  under  the  greater, 
so  that  figures  of  the  same  order  stand  in  the  same  column. 

2.  Subtract  as  in  simple  numbers,  and  place  the  decimal 
point  in  its  proper  place  in  the  difference. 

2.  From  63.72  take  25.81. 

3.  From  96.32  take  73.15. 

4.  From  123.16  take  75.84. 

5.  From  247.125  take  167.183. 

6.  From  1  and  1  tenth  take  1  tenth  and  1  thousandth. 

7.  From  2  and  2  hundredths  take  2  tenths  and  2 
thousandths. 

8.  From  3  tenths  take  3  ten-thousandths. 

9.  From  7  take  7  tenths  and  707  millionths. 


DECIMAL    FRACTIONS.  119 


MULTIPLICATION  OF  DECIMALS. 

135.  Multiplication  of  Decimals  is  the  process  of 
multiplying  when  one  or  both  terms  are  decimals. 

1.  Multiply  7.23  by  .46. 

Solution  1. — Multiplying  as  in  whole  num-  operation. 

bers,  we  have  33258;  now,  if  the  multiplicand  7.23 

alone  were  hundredths,  the  product  would  be  one-  .46 

hundredth  of  this,  or  332.58 ;  but  since  the  mul-  4303 

tiplier  is  also  hundredths,  the  product  is  one-hun-  2892 

dredth  of  332.58,  which,  by  moving  the  decimal  3.8258 
point  two  places  to  the  left,  becomes  3.3258. 

Solution  2.-7.23  X  -46  =  HfX^  =  -^  =to!ooX  -258 

10000 

=  3.3258.     From  either  of  these  solutions  we  derive  the  following 

Rule. — Multiply  as  in  loliole  numbers,  and  point  off  as 
many  decimal  places  in  the  product  as  there  are  in  both  mul- 
tiplier and  multiplicand,  prefixing  ciphers  when  necessary. 

2.  Multiply  15.17  by  .IS. 

3.  Multiply  26.18  by  .25. 

4.  Multiply  53.46  by  .35. 

5.  Multiply  67.38  by  1.26. 

6.  Multiply  138.25  by  2.47. 

7.  Multiply  466.72  by  5.29. 

8.  Multiply  407.03  by  7.35. 

9.  Multiply  620.75  by  12.36. 

10.  Multiply  725.82  by  23.08. 

11.  Multiply  .00723  by  .0317. 

12.  Multiply  1.0309  by  .00321. 

DIVISION  OF  DECIMALS. 

136.  Division  of  Decimals  is  the  process  of  dividing 
when  one  or  both  terms  are  decimals. 

1.  Divide  7.8315  by  2.27. 


120  DECIMAL   FRACTIONS. 

Solution. — If  we  divide  as  in  whole  num-  operation. 

bers,  we  obtain  a  quotient  of  345  ;  now,  since  2.27)7.8315(3.45  Ana 
the  dividend  is  the  product  of  the  divisor  and  c  81 

quotient,  the  number  of  decimal  places  in  the  1  021 

dividend  must  equal  the  number  in  the  divisor  908 

and  quotient:  hence,  the  number  of  decimal  1135 

-i  -i  or 

places  in  the  quotient  must  equal  the  number  Xl0° 

of  decimal  places  in  the  dividend  diminished 

by  the  number  in  the  divisor  ;  hence,  there  should  he  four  minus  two, 

or  tioo  decimal  places  in  the  quotient,  therefore  the  quotient  is  3.45. 

Q  n  7   QQ1  K      .      9   97    78315    _i_    227    78  315    V    W  ■ 

C50LLTI0N    Z.  —  i.OoLO  —  £—t    —  TO  0  0  0  ±($0   TOO  0  0    A   Hz7   — 

78315    =  T^  x  7f  I?- 5  =  jh  X  3^5  =  3-45-  From  either  of  these 

100^227 

solutions  we  derive  the  following 

Rule. — Divide  as  in  whole  numbers,  and  point  off  as 
many  decimal  places  in  the  quotient  as  the  number  of  deci- 
mal places  in  the  dividend  exceeds  the  number  in  the  divisor. 

Note  1. — When  there  are  not  as  many  decimal  places  in  the  divi- 
dend as  in  the  divisor,  annex  ciphers  to  make  the  number  of  place? 
equal. 

2.  When  the  number  of  figures  in  the  quotient  is  less  than  tho 
excess  of  the  decimal  places  in  the  dividend  over  those  in  the  divisor, 
ciphers  must  be  prefixed  to  the  quotient. 

2.  Divide  14.1372  by  4.5.  Ans.  3.1416. 

3.  Divide  196.1875  by  10.75.  Ans.  18.25. 

4.  Divide  25.1328  by  8.  Ans.  3.1416. 

5.  Divide  65.9736  by  3.1416. 

6.  Divide  2450.448  by  .5236. 

7.  Divide  2748.9  by  .7854. 

8.  Divide  127.328  by  .07958. 

9.  Divide  15.90435  by  20.25. 
10.  Divide  352.0625  by  32.75. 

PRACTICAL   PROBLEMS. 

1.  What  cost  43.45  acres  of  land  at  $38.5  an  acre? 

Ans.  $1672.825. 

2.  "What  cost  57.75  tons  of  hay  at  $12.25  a  ton  ? 

Ans.  $707.4375. 


DECIMAL    FRACTIONS.  121 

3.  If  31.25  yards  of  muslin  cost  $7.8125;  how  much 
is  that  a  yard  ?  Ans.  $0.25. 

4.  A  man  sold  35.25  pounds  of  butter  for  $5,875  ;  how 
much  is  that  a  pound?  Ans.  $0,166+. 

5.  There  are  7.92  inches  in  a  link  ;  how  many  inches 
in  990  links  ?  Ans.  7840.8  in. 

6.  There  are  31.5  gallons  in  a  barrel ;  how  many 
barrels  in  2756.25  gallons  ?  Ans.  87.5  barrels. 

.     7.  If  14.5  yards  of  cloth  cost  $68,875,  how  much  is 
that  a  yard  ?  Ans.  $4.75. 

8.  If  a  man  walk  112-1184  miles  in  9.16  days,  how 
many  miles  does  he  walk  each  day  ?     Ans.  12.24  miles. 

9.  How  many  yards  of  cloth  at  $4.28  a  yard,  can  a 
person  buy  for  $44.9828  ?  Ans.  $10.51. 

10.  What  is  the  value  of  54.6  multiplied  by  80.5,  and 
the  product  divided  by  2  ?  Ans.  2197.65. 

11.  The  circumference  of  a  water-wheel  is  64  feet,  and 
the  diameter  equals  this  divided  by  3.1416;  required 
the  diameter  ?  Ans.  20.3718  feet. 

12.  If  25.5  yards  of  cloth  cost  195.375,  how  much  will 
45.25  yards  cost  ?  Ans.  $346,696+. 

13.  If  an  imperial  gallon  contains  277.274  cubic  inches, 
how  many  cubic  inches  in  328.55  gallons? 

Ans.  91098.3727. 

14.  A  gallon  of  distilled  water  weighs  8.33888  pounds; 
how  many  gallons  in  1000  pounds  of  such  water  ? 

Ans.  119.92+. 

15.  A  cubic  inch  of  water  weighs  252.458  grains;  how 
many  cubic  inches  in  157786.25  grains  ? 

Ans.  625cu.  in. 

16.  A  drew  41.25  barrels,  of  31.5  gallons  each,  from  a 
cistern  containing  2000  gallons;  how  much  remained? 

Ans.  700.625. 

17.  A  bought  7S.25  acres  of  land  at  $128.5  an  acre, 
and  sold  it  fur  $97^1.25;  what  was  the  loss  on  each 
acre?  Ans.  $3.50. 

11 


122  PROBLEMS. 

REVIEW   OF   FUNDAMENTAL   RULES. 

HISTORICAL,  GEOGRAPHICAL,  ETC.  PROBLEMS. 
Suggestion. — The  teacher  should  explain  the  nature  of  the  facts 
presented,  and  require  the  pupils  to  remernher  some  of  the  mora 
important  numbers  and  dates. 

PROBLEMS 

on  Battles  of  the  Revolution. 

1.  At  the  battle  of  Lexington,  the  Americans  lost  90 
men,  the  British  190  more ;  how  many  did  the  British 
Jose? 

2.  At  the  battle  of  Bunker  Hill,  the  Americans  had 
1500  men,  the  British  1500  more;  how  many  had  the 
British  ? 

3.  In  this  battle  the  Americans  lost  450  men,  the 
British  604  more;  how  many  did  the  British  lose? 

4.  At  the  battle  of  Long  Island,  the  British  lost  367 
men,  the  Americans  1233  more;  how  many  did  the 
Americans  lose  ? 

5.  At  the  battle  of  Trenton,  the  British  lost  45  in 
killed  and  wounded,  and  1000  prisoners;  what  was 
their  loss  ? 

6.  In  the  battle  of  Brandywine,  the  British  lost  800 
men,  and  the  Americans  450  more ;  how  many  did  the 
Americans  lose  ? 

7.  In  the  battle  of  Germantown,  the  British  lost  600 
men,  and  the  Americans  lost  600  more ;  how  many  did 
the  Americans  lose  ? 

8.  At  the  battle  of  Bennington,  the  Americans  lost 
about  100,  and  the  British  600  more;  required  the 
British  loss. 

9.  At  the  battle  of  Monmouth,  the  Americans  lost  70 
ii?  killed,  and  the  British  230  more;  required  the  British 
loss. 

10.  In  taking  Stony  Point,  Gen.  Wayne  lost  15  killed 
and  83  wounded,  and  the  British  lost  500  more  in  killed, 
wounded,  and  prisoners ;  required  the  British  loss. 


PROBLEMS.  123 

11.  At  the  battle  of  Sander's  Creek,  the  British  lost 
325,  and  the  Americans  675  more;  what  was  the  Ame- 
rican loss? 

12.  At  the  battle  of  Kind's  Mountain,  the  Americans 
lost  20  men,  and  the  British  280  more;  how  many  did 
the  British  lose  ? 

13.  At  the  battle  of 'Guilford,  the  Americans  lost  400 
men,  the  British  lost  100  more;  required  the  British 
loss. 

14.  At  Hobkirk's  Hill,  the  British  loss  was  about  258 
men,  and  the  Americans  8  more;  how  many  did  the 
latter  lose? 

15.  At  Xinety-Six,  the  Americans  lost  51  in  killed 
and  Avounded ;  the  British  lost  1  more  than  this  in 
killed,  and  283  more  in  wounded;  required  the  British 
loss. 

1G.  At  the  battle  of  Eutaw  Springs,  the  Americans 
lost  555,  and  the  British  138  more;  required  the  British 
loss. 

17.  At  Yorktown,  Washington  had  11,000  Americans 
and  5000  French,  and  the  British  had  2000  more  than 
the  French  ;  what  was  the  force  on  each  side  ? 

18.  At  Yorktown,  the  Americans  lost  about  75  killed, 
and  225  wounded ;  the  British  lost  156  killed,  170  more 
than  this  wounded,  and  70  missing ;  what  wTas  the  loss 
on  each  side,  not  including  prisoners  ? 

PROBLEMS    IN    AMERICAN    HISTORY. 

1.  America  was  discovered  by  Columbus  in  1492,  and 
Jamestown  was  settled  in  1607;  wThat  was  the  differ- 
ence of  time? 

2.  Plymouth  was  settled  in  1620 ;  how  long  was  that 
after  America  was  discovered,  and  how  long  after  the 
settlement  of  Jamestown? 

3.  The  battle  of  Lexington  was  fought  in  1775;  how 
long  was  that  after  the  settlement  of  Plymouth? 

4.  The   .Declaration  of    Independence  was   made   in 


124  PROBLEMS. 

1776  ;  how  long  was  that  after  the  settlement  of  James, 
town  ? 

5.  The  surrender  of  Burgoyne  took  place  in  1777  j 
how  long  was  that  after  the  discovery  of  America? 

6.  The  Inauguration  of  Washington  took  place  iL 
1789;  how  long  was  that  after  the  battle  of  Bunker 
Hill,  in  1775  ? 

7.  The  battle  of  New  Orleans  took  place  in  1815; 
how  long  was  that  after  the  inauguration  of  Washing- 
ton ? 

8.  The  frigate  Constitution  captured  the  British  fri- 
gate Guerriere  in  1812;  how  long  was  it  after  the  De- 
claration of  Independence? 

9.  Commodore  Perry  won  his  great  naval  victory  in 
1813 ;  how  long  was  that  after  the  battle  of  Lexington? 

10.  General  Jackson  won  his  great  victory  at  New 
Orleans  in  1815 ;  how  long  is  it  from  then  till  the  pre- 
sent ? 

11..  General  Packenham  had  12000  men,  and  General 
Jackson  6000 ;  how  many  more  had  the  British  J/;han 
the  Americans  ? 

12.  The  British  lost  1700  in  killed  and  wounded,  the 
Americans  13  men;  what  was  the  difference? 

13.  War  with  Mexico  commenced  in  1846 ;  how  long 
was  that  after  the  battle  of  Lexington  ? 

14.  At  the  battle  of  Palo  Alto,  General  Taylor  had 
2300  men  and  the  Mexicans  6000 ;  what  was  the  differ- 
ence? 

15.  The  battle  of  Buena  Yista  was  fought  in  1847 ; 
how  long  was  this  after  the  battle  of  Bunker  Hill  ? 

16.  At  this  battle  General  Taylor  had  4759  men,  while 
Santa  Anna  had  20000  men ;  required  the  difference  of 
the  forces. 

17.  General  Scott  took  the  Mexican  capital  in  1847; 
how  long  is  it  from  that  time  to  the  present  ? 


PROBLEMS.  125 

PROBLEMS  ON  THE  AREA  OF  STATES. 
NEW  ENGLAND  STATES. 

1.  The  area  of  Maine  is  30000  square  miles,  and  of 
jjSew  Hampshire  9280  square  miles;  how  mueh  larger 
is  the  former? 

2.  Vermont  contains  9056  square  miles,  and  Massa- 
chusetts 7800  square  miles;  how  much  larger  is  the 
former  than  the  latter? 

3.  Ehode  Island  contains  1306  square  miles,  and  Con- 
necticut 4674  square  miles;  how  much  larger  is  Maine 
than  both  of  these? 

4.  Which  is  larger,  and  how  much,  Maine  or  all  the 
rest  of  the  .New  England  States  ?  Which  is  larger,  and 
how  much,  New  Hampshire  and  Vermont  together,  or 
Massachusetts  and  Connecticut  together  ? 

MIDDLE   STATES. 

5.  New  York  contains  47000  square  miles,  and  New 
Jersey  8320  square  miles;  how  much  larger  is  the  former? 

6.  Pennsylvania  contains  46000  square  miles,  and 
Delaware  2120  square  miles;  how  much  larger  is  Penn. 
sylvania  ? 

7.  Maryland  contains  9356  square  miles;  how  much 
larger  is  Maryland  than  New  Jersey  ? 

8.  How  much  larger  is  Pennsylvania  than  New  Jer- 
sey,  Delaware,  and  Maryland  all  together? 

9.  How  much  larger  are  the  Middle  States  than  the 
New  England  States? 

WESTERN   STATES. 

10.  Ohio  contains  39964  square  miles,  and  Indiana 
33809  square  miles;  how  much  larger  is  the  former 
than  the  latter  ? 

11.  Michigan  contains  56243  square  miles,  and  Illinois 
55405  square  miles;  how  much  larger  is  the  formed 
than  the  latter  ?  n* 


126  PROBLEMS. 

12.  How  much  larger  are  Ohio  and  Michigan  than 
Indiana  and  Illinois? 

13.  Wisconsin  contains  53924  square  miles,  and  Iowa 
55045  square  miles;  how  much  larger  is  the  latter  than 
the  former  ? 

14.  Missouri  contains  67380  square  miles,  and  Ken- 
tucky 37680  square  miles;  how  much  larger  is  Mis- 
souri ? 

15.  Which  would  make  the  larger  State,  Wisconsin 
and  Iowa,  or  Missouri  and  Kentucky? 

16.  California  contains  189000  square  miles,  and  Ore- 
gon 95000  square  miles;  which  is  the  larger,  and  how 
much  ? 

17.  How  much  larger  are  the  Western  States  than 
the  New  England  and  Middle  States  together  ? 

SOUTHERN   STATES. 

18.  "Virginia  contains  41352  square  miles,  and  West 
Virginia  20000  square  miles;  how  much  larger  is  Vir- 
ginia than  West  Virginia  ? 

19.  North  Carolina  contains  45000  square  miles,  and 
South  Carolina  24500  square  miles;  how  much  larger 
is  North  Carolina  than  South  Carolina  ? 

20.  Georgia  contains  58000  square  miles,  and  Louisiana 
46431  square  miles ;  how  much  larger  is  G-eorgia  than 
Louisiana? 

21.  Alabama  contains  50722  square  miles,  and  Missis- 
sippi 47156  square  miles ;  how  much  larger  is  Alabama 
than  Mississippi  ? 

22.  Arkansas  contains  52198  square  miles,  and  Tennes- 
Bee  45600  ;  which  is  the  larger,  and  how  much  ? 

23.  Florida  contains  59628  square  miles,  and  Texas 
237321  square  miles ;  how  much  larger  is  Texas  than 
Florida  ? 

24.  Which  is  larger,  and  how  much,  Texas,  or  all  the 
other  States  taken  together  ? 


BUSINESS    PROBLEMS.  127 

BUSINESS    PROBLEMS. 

Suggestion. — Pupils  will  put  these  in  the  form  of  accounts,  as  on 

page  93. 

1.  A  merchant  sold  a  farmer  125  yards  of  calico,  at 
18  cents  a  yard,  150  yards  of  drilling,  at  15  cents  a  yard, 
and  bought  of  the  farmer  225  bushels  of  oats,  at  40  cents 
a  bushel,  and  90  bushels  of  rye,  at  $1.25  a  bushel ;  which 
owes  the  other,  and  how  much? 

2.  A  mechanic  sold  a  farmer  a  wagon  for  $56.50,  two 
plows,  at  $7.50  each,  and  6  wheel-barrows,  at  $5.25  each; 
and  bought  of  the  farmer  50  bushels  of  potatoes,  at  75 
cents  a  bushel,  and  75  bushels  of  wheat,  at  85  cents  a 
bushel ;  which  owes  the  other,  and  how  much  ? 

3.  A  farmer  sold  a  merchant  4  cows,  at  $28.50  each,  a 
yoke  of  oxen  for  $95,  and  7  sheep,  at  $6.25  each;  and 
took  in  payment  40  yards  of  carpet,  at  $2.25  a  yard,  35 
yards  of  cloth,  at  $3.25  a  yard,  and  58  yards  of  muslin, 
at  15  cents  a  yard;  how  much  remains  due? 

4.  A  farmer  bought  of  a  mechanic,  2  wagons,  at  $76 
each,  4  drags,  at  $6.50  each,  3  harrows,  at  $12.25  each; 
and  sold  him  45  bushels  of  apples,  at  55  cents  a  bushel, 
3  barrels  of  cider,  at  $5.25  a  barrel,  28  bushels  of  corn, 
at  42  cents  a  bushel,  and  3  cows,  at  $28.75  each;  which 
owes  the  other,  and  how  much  ? 

5.  A  mechanic  bought  of  a  merchant 

28  pounds  of  sugar,  at  18cts.  a  pound, 

36  pounds  of  rice,  at  17cts.  a  pound, 
45  yards  of  muslin,  at  18cts.  a  yard, 
28  yards  of  cloth,  at  $5.25  a  yard, 

37  barrels  of  flour,  at  $7.25  a  barrel ; 
And  sold  him 

4  wagons,  at  $75  each, 

6  wagon-racks,  at  $13.50  each, 

2  mowing-machines,  at  $157  each, 

3  ox-yokes,  at  $6.75  each ; 
Which  owes  the  other,  and  how  much? 


128  DENOMINATE    NUMBERS. 

SECTION"   VII. 

DENOMINATE  NUMBERS. 

137.  A  Concrete  Number  is  one  which  refers  to 
some  particular  unit,  as  2  books,  3  pounds,  etc. 

138.  Concrete  numbers  are  of  two  kinds ;  those  in 
which  the  unit  is  natural,  and  those  in  which  it  is  arti- 
ficial. 

139.  A  Denominate  Number  is  a  concrete  number 
in  which  the  unit  is  artificial,  as  3  pounds,  4  yards,  5 
minutes,  etc. 

14 0.  Reduction  is  the  process  of  changing  a  number 
from  one  denomination  to  another  without  changing  its 
value. 

141.  Reduction  Descending  is  the  process  of  re- 
ducing from  a  higher  to  a  lower  denomination. 

142.  Reduction  Ascending  is  the  process  of  re- 
ducing from  a  lower  to  a  higher  denomination. 

ENGLISH  MONEY. 

143.  English,  or  Sterling  Money,  is  the  money  of 
England. 

TABLE. 

4  farthings  (far.,  or  qr.)  equal  1  penny,     .     .     .     d. 

12  pence "      1  shilling,  .     .     .     s. 

20  shillings "      1  pound,*  .     .     .     £. 


21  shillings "      1  guinea. 


to' 

Sb *  s"^-1^""  s* 

Mental  Exercise;* — Repeat  the  table  of  English  Money. 

How  many  far.  in  2d.  ?  in  3d.  ?  in  Gd.  ?  in  8d.  ? 

How  many  pence  in  I2far.  ?  in  lGfar.  ?  in  20far.  ?  in  28far.  ?, 

How  many  pence  in  2s.  ?  in  3s.  ?  in  5s.  ?  in  6s.  ? 

How  many  far.  in  Is.  ?  in  2s.  ?  in  3s.  ?  in  5s.  ? 


*  The  £  coined  in  gold  is  called  a  sovereign.  Its  value  is  $4.84.  A  five- 
shilling  piece  in  silver  is  called  a  crown.  A  two-and-a-half-shilling  pieo* 
in  silver  is  called  a  half-Town. 


DENOMINATE    NUMBERS.  129 


REDUCTION  DESCENDING. 
1.  How  many  farthings  in  8  pence  and  3  farthings? 


8 
4 


Solution. — In  one  penny  there  are  4  operation. 

farthings,  hence  4  times  the  number  of  d.     far. 

pence  equal  the  number  of  farthings;  4 
times  8  are  32,  and  32far.  plus  the  3  far. 
equal    35  farthings.  32 

3 

3  ofar.  Ans. 

2.  How  many  farthings  in  14d.  2far.?     Ans.  58far. 

3.  How  many  pence  in  15s.  9d.  ?  Ans.  189d. 

4.  How  many  shillings  in  £23  10s.?  Ans.  470s. 

5.  How  many  farthings  in  23s.  lOd.  3far.  ? 

6.  How  many  pence  in  £32  19s.  3d.  ? 

REDUCTION  ASCENDING. 

1.  How  many  shillings,  pence,  and  farthings,  in  1487 
farthings  ? 

Solution. — There  are  4  farthings  in  one  operation. 

penny,  hence  in   1487far.  there    are    as  4)1487 

many  pence  as  4  is  contained  times  in  12)371-3far. 

1487,   which  are    371    pence,   and    3far.  30-lld. 

remaining.     There  are   12  pence  in  one 

shilling,  and  in  371  pence  there  are  as  many  shillings  as  12  is  con- 
tained times  in  371,  which  are  30  shillings,  and  11  pence  remaining. 
Hence,  1485far.  equal    30s.,  lid.,  3far. 

2.  How  many  shillings,  pence,  and  farthings  in  989 
farthings  ?  Ans.  20s.  7d.  lfar. 

3.  Reduce  2676  farthings  to  shillings  and  pence. 

Ans.  55s.  9d. 

4.  How  many  pounds  in  3178  farthings  ? 

Ans.  £3  6s.  2d.  ?far, 

5.  How  many  pounds  in  9761  pence? 

6.  How  many  guineas  in  17654  farthings  ? 


loO 


DENOMINATE    NUMBERS. 


TROY  WEIGHT. 
144.   Troy  Weight  is  used  in  weighing  gold,  silver, 
jewels,  etc. 

TABLE. 

24  grains  (gi\)     .      equal  1  pennyweight, .     .     pwt. 
20  pennyweights      .     "      1  ounce,     .     .     .     .     oz. 
12  ounces    .     .     .     .     "      1  pound,     ....     lb. 


Mental  Exercises. — How  many  oz.  in  21b.  ?  in  31b.  ?  in  51b.  ? 
How  many  \\.  in  3Goz.  ?  in  48oz.  ?  in  60oz.  ?  in  lpwt.  ?  in  2pwt.  ? 
How  many  pwt.  in  2oz.  ?  in  3oz.  ?  in  4oz.  ?  in  48grs.  ?  in  72grs.  ? 


1.  How  many  grains  in 
15oz.  16pwt.  13grs.? 
oz.      pwt. 
15      16 

20 


grs. 

13 


300 
'16 


316  pwt. 
24 


2.  How  many  penny- 
weights in  74597grs.? 

grs. 
24)7597 

2|0)31|6  —  13grs. 
15  —  16pwt. 
Ans.  15oz.  16pwt.  13grs. 

Note. — In  dividing  by  large 
numbers,  like  24,  we,  of  course, 
divide  by  Long  Division.  We  in- 
dicated the  result  above. 


1264 
632 

7584 
13 

7597grs.  Ans. 

How  many 

3.  Grains  in  6pwt.  12grs.  ? 

4.  Pounds,  oz.,  and  pwts.  in  963pwts.  ? 

Ans.  48oz.  3pwt 

5.  Grains  in  3oz.  llpwt.  14gr.  ? 

6.  Oz.,  pwt.,  and  grs.  in  5170grs.  ? 

7.  Pounds,  oz.  etc.,  in  15786grs.  ? 


Ans.  156grs. 


DENOMINATE    NUMBERS. 


131 


APOTHECARIES  WEIGHT.. 

145.  Apothecaries  Weight  is  used  in  mixing  medi- 
cines. Medicines  are  bought  and  sold  by  Avoirdupois 
Weight. 

TABLE. 


20  grains  (gr.) 
3  scruples 
8  drams    .     . 

12  ounces    . 


equal  1  scruple, .     .     .     .  9 . 

1  dram,     ....  3. 

1  ounce,    .     .     .     .  3. 

1  pound,  ....  lb. 


a 
it 

u 


Mental  Exercises. — 1.  How  many  grs.  in  2  scruples?  in  3  scru- 
ples ?  in  4  scruples. 

2.  How  many  scruples  in  40grs.  ?  in  GOgrs.  ?  in  80grs.  ?  inl60grs.? 

3.  How  many  scruples  in  2  drams?  in  4  drams?  in  2  ounces?  in 
4  ounces  ? 

4.  How  many  drams  in  2  ounces?  inlft)?  in!25?in365?  in!20grs.? 


1.  How  many  grains  in 
153  2B  12gr.  ?  ' 


OrERATION. 

15     2     12 
3 

47 

20 

952gr.  Ans. 

Note. — "When  convenient,  add 
in  the  numbers  as  we  multiply. 


How  many 

1.  Drams  in  71b.  53  ?  Ans.  7123. 

2.  Pounds  and  ounces  in  2305  ?  Ans-  19tb-  115- 

3.  Scruples  in  191b.  85  53  2£  ?  Ans.  5681B- 

4.  Pounds,  etc.,  in  92375o;r.?     Ans.  161b.  85  I3  15gr- 


2.  How  many  drams  in 
952  grains? 

OPERATION. 

2(0)952 

3)47^ 12gr. 

15  —  29 
Ans.  153  29  12gr 


? 


132 


DENOMINATE    NUMBERS. 


AVOIRDUPOIS  WEIGHT. 

146.  Avoirdupois  Weight  is  used  for  weighing  every 
thing  except  jewels,  precious  metals,  etc. 


TABLE. 

16  drams  (dr.)    .  equal   1  ounce,  oz. 

16  ounces  .       "       1  pound,  lb. 

25  pounds       .     .       u       1  quarter,  qr. 

4  quarters     .     .       "       1  hundred-weight,  cwt. 
20  hundred-weight  "      1  ton,  T. 

Mental  Exercises. — -Ask  mental  questions  upon  this  and  the 
following  tables,  similar  to  those  suggested  under  the  previous 
tables. 


N 


)w  many  drams  in 

2.   How   many   poundf 

oz.  9dr  ? 

in  3289  drams?" 

OPERATION. 

OPERATION. 

lb.      oz.    dr. 

drams. 

12     13     9 

16)3289 

16 

16)  205  —  9dr. 

72 

12  —  13oz. 

12 

Ans.  121b.  13oz.  9dr. 

13 

205  oz.,  etc. 

How  many 
3    Drams  in  171b.  12oz.  lldr.  ? 

4.  Ounces  in  3qr  151b.  13oz.  ? 

5.  Pounds  in  5cwt.  3qr.  161b.  ? 

6.  Pounds  in  15675  drams  ? 

7.  Quarters  in  27392  ounces  ? 

8.  Drams  in  20cwt.  3qr.  141b.  lloz.  9dr.  ? 
9    Hundred-weight  in  17896754  drams  ? 


DENOMINATE    NUMBERS.  13S 


WINE  MEASURE. 

147.   Wine  Measure  is  used  for  measuring  nearly 
all  kinds  of  liquids. 

TABLE. 

4  gills  (gi.)  ....    equal  1  pint,        pt. 

2  pints "       1  quart,      qt. 

4  quarts "       1  gallon,    gal. 

Note. — The  wine  gallon  contains  231  cubic  inches,  while  the  beer  gallon, 
used  in  measuring  beer,  and  sometimes  milk,  contains  281  cu.  in.  In  the 
old  tables  were  given  3U  gals.  =  1  barrel;  63  gals.  =  1  hogshead:  2 
hogsheads  =  1  pipe ;  2  pipes  =  1  tun.  These  are  not  measures,  however, 
but  vessels  of  variable  capacity. 

How  many 

1.  Pints  in  4gal.  3qt.  lpt.? 

2.  Gallons  in  976  pints  ? 

3.  Gills  in  17gal.  2qt.  3gi.  ? 

4.  Gallons  in  1763  gills  ? 


DRY  MEASURE. 

148.  Dry  Measure  is  used  in  measuring  dry  sub- 
stances, as  grain,  fruit,  salt,  coal,  etc. 

TABLE. 

2  pints  (pt.)     .     .     .     equal  1  quart,      qt. 
8  quarts       ....         "1  peck,        pk. 
4  pecks "1  bushel,    bu. 

How  many 

1.  Pints  in  3pk.  6qt.  lpt.  of  berries? 

2.  Bushels  in  314  quarts  of  clover  seed? 

3.  Bushels  in  3157  pints  of  cranberries  ? 

4.  What  cost  .4  bushels  of  berries  at  2  cents  a  pint  f 


134  DENOMINATE    NUMBERS. 

APOTHECARIES'  FLUID  MEASURE. 

149.  Apothecaries'  Fluid  Measure  is  used  for  mea- 
suring liquids  in  preparing  medical  prescriptions. 

TABLE. 

60  minims  (Tfp)     .  equal  1  nuidrachm,  f^. 

8  fluidrachms      .        "      1  fluidounce,  fg. 
16  fluidounces       .        "      1  pint, '  O. 

8  pints  ....        "1  gallon,  Cong. 

^"0TE. — o  is  the  initial  of  octans,  the  Latin  for  one-eighth,  the  pint  being 
|  of  a  gallon.  Cong,  is  the  abbreviation  of  congiarium,  the  Latin  fol 
gailon. 

How  many 

1.  Minims  in  20.  5f^  ? 

2.  Pints  in  8000  minims  ? 

3.  Fluidounces  in  BCong.  70.  5f  J  ? 

4.  Gallons  in  78561  minims  ? 

MEASURE  OF  LENGTH. 

150.  Measure  of  Length,  or  Long  Measure,  is  used 
for  measuring  length,  breadth,  height,  distances,  etc. 

1.  A  Line  is  that  which  has  length 

Without  breadth  or  thickness.  .        /D 

2.  An  Angle  is  the  opening  between  j    / 

two  lines  which  diverge  from  a  common  A g B 

point.     Thus,  ACD  and  DCB  are  angles. 

3.  A  Right  Angle  is  formed  by  one  line  perpendicular  to 
another,  as  ACE  or  ECB. 

TABLE. 

12  inches  (in.)    . 


3  feet    . 
5  A  yards 
40  rods  . 


8  furlongs 


equal  1  foot, 

ft. 

"      1  yard, 

yd. 

"      1  rod, 

rd. 

"      1  furlong, 

fur. 

"      1  mile, 

mi. 

.  ■  Note.—  Cloth  Measure  is  not  now  used.     Cloth,  muslin,  etc.  are  bought 
by  the  yard,  half-yard,  eighth,  etc. 


DENOMINATE    NUMBERS. 


135 


1.  How   many    feet    in 
I2rd.  3yd.  2ft.  ? 

OPERATION. 

rd.    yd.    ft. 

12     3     2 
_5| 

63 
6 


69yd. 
3 


209ft.  Ans. 

Note. — We  multiply  by  5,  and  add 
to  the  product  the  3  yds.,  and  then 
multiplying  by  £,  we  have  69  yd. 


2.  How   many   rods   in 
209  ft,  ? 

OPERATION, 
feet. 
3)209 

5*)69ft, 

2       2 


11)138 

12—  6halves=3yi 
Ans.  12rd.  3yd.  2ft. 

Note. — To  divide  by  b\,  we  re- 
duce both  to  halves,  then  the  re- 
mainder is  halves,  which  we  reduce 
to  wholes,  by  dividing  by  2. 


Ans.  281ft. 

Ans.  635in. 

Ans.  87yds.  etc. 


How  many 

3.  Feet  in  16rd.  5yd.  2ft.  ? 

4.  Inches  in  17yd.  1ft.  llin.  ? 

5.  Yards  in  3146  inches? 

6.  Eods  in  6547  inches  ? 

7.  Feet  in  7fur.  32rd.  4yd.  ? 

8.  Furlongs  in  4389  feet? 

9.  Miles  in  19280  feet? 

10.  Inches  in  2m.  6fur.  4rd.  8in.  ? 

11.  How  many  inches  from  New  York  to  Philadelphia, 
if  the  distance  is  96  miles  ? 


SURFACE  OR  SQUARE  MEASURE. 

151.  Surface  or  Square   Measure  is  used  in  mea- 
suring surfaces,  as  land,  boards,  etc. 

1.  A  Surface  is  that  which  has  length  and  breadth  -without 
thickness. 

2.  A  Square  is  a  surface  which  has  four  equal 
sides  and  four  right  angles,  as  in  the  margin. 


136 


DENOMINATE    NUMBERS. 


3.  A  Rectangle  is  a  surface  which  has  four 
sides  and  four  right  angles.  A  slate,  a  door, 
the  sides  of  a  room,  etc.,  are  examples  of 
rectangles. 

4.  The  Area  of  a  surface  is  expressed  by  the  number  of  times 
it  contains  a  small  square  as  aunit  of  measure. 

5.  The  area  of  a  square  or  rectangle  is  equal  to  the  length  multi' 
plied  by  the  breadth.  For,  in  the  rectangle  above,  the  whole 
number  of  little  squares  is  equal  to  the  number  in  each  row 
multiplied  by  the  number  of  rows :  that  is,  4  X  3  which  equals 
12,  which  is  the  same  as  the  number  of  units  in  length  multiplied 
by  the  number  in  breadth. 


TABLE 

144  square  inches  (sq.  in.)  equal  1  square  foot, 


9  square  feet  .  . 
30 \  square  yards  . 
40  perches    .     .     . 

4  roods  .  .  i  . 
640  acres  .     .     .     . 


1.  How  many  square 
feet  in  28P.  18sq.  yd.  5 
sq.  ft.  ? 

OPERATION. 
P.     sq.  yd.    sq.  ft. 
28       18        5 

30* 


865sq.  yd. 
9 

7790sq.  ft. 

Note. — We  multiplied  by  30, 
added  in  the  18  sq.  yds.,  and  then 
multiplied  by  i  and  took  the  sum. 


sq.  ft. 
1  square  yard,         sq.  yd. 
1  perch,  or  sq.  rod,  P. 
1  rood,  R. 

1  acre,  A. 

1  square  mile,         sq.  mi. 

2.   How   many   perches 
are  there  in  7790sq.  ft.  ? 

OPERATION 
sq.  ft. 

9)7790 

30^)865  —  5sq.  ft. 
4        4 


121)3460 

28  —  72  fourths, 
or  18sq.  yd. 
Ans.  27P.  18sq.  yd.  5sq.  ft. 

Note. — To  divide  by  30J  we  re- 
duce  both  divisor  and  dividend  to 
4ths,  and  then  divide ;  the  remain- 
der ie  72  fourths,  or  18  sq.  yd. 


DENOMINATE    NUMBERS. 


137 


How  many 

3.  Square  inches  in  2sq.  yd.  3sq.  ft.  ? 

4.  Square  feet  in  2R.  13P.  16sq.  yd.  ? 

5.  Perches  in  8765  square  feet  ? 

6.  Acres  in  1997  perches  ?  Ans.  12A.  IE.  37P. 

7.  Perches  in  29A.  3R.  19P.?  Ans.  4779P. 

8.  Acres  in  89763  square  yards  ? 

Ans.  18A.  2E.  7P.,  etc. 


3  feet  long. 


CUBIC  OK  SOLID  MEASURE. 

152.  Cubic  or  Solid  Measure  is  used  in  measuring 
things  which  have  length,  breadth,  and  thickness. 

1.  A  Volume  is  that  which  has 
length,  breadth,  and  thickness.  A 
volume  is  also  called  a  solid. 

2.  A  Cube  is  a  volume  bounded 
by  six  equal  squares.  A  Rect- 
angular Volume  is  one  bounded 
by  rectangles.  Cellars,  boxes,  rooms, 
etc.,  are  examples  of  rectangular 
volumes. 

3.  The  Contents  of  a  volume  are  expressed  by  the  number  of 
times  it  contains  a  cube  as  a  unit  of  measure. 

The  contents  of  a  Cube  or  Rectangular  Solid  are  equal  to  the  pro- 
duct of  the  length,  breadth,  and  height. 

Fur  in  the  volume  above,  the  number  of  cubic  units  on  the  base 
is  equal  to  the  length  multiplied  by  the  breadth,  and  the  whole 
number  of  cubic  units  equals  the  number  on  the  base  multiplied 
by  the  number  of  layers  ;  hence  the  whole  number  equals  3  X 
3X3  =  27. 

4.  A  cord  of  wood  is  a  pile  8 
feet  long,  4  feet  wide,  and  4  feet 
high.  A  cord  foot  is  a  part  of 
this  pile,  1  foot  long;  it  equals 
10  cubic  feet. 

12* 


1  cord. 


138 


DENOMINATE    NITOIBERS. 


TABLE. 

1728  cubic  inches  (cu.  in.)  equal  1  cubic  foot, 
27  cubic  feet     .     .     . 
16  cubic  feet     .     .     . 
8  cord  feet,  or  "| 
128  cubic  feet,     j 
40  feet  of  round  timber,  or 
50  feet  of  square  timber 


u 


u 


ii 


,  or  I 


cu.  ft. 
1  cubic  yard*  cu.  yd 
1  cord  foot,  cd.  ft. 

1  cord  of  wood,  Cd. 
1  ton,  tn. 


How  many 

1.  Cu.  in.  in  7cu.  ft.  96cu.  in.  ? 

2.  Cu.  in.  in  12cu.  yd.  25cu.  ft.? 

3.  Cu.  ft.  in  8469  cubic  inches? 

4.  Cu.  yd.  in  60463  cubic  inches? 

5.  Cords  in  8192  cubic  feet? 


MEASURE  OF  TIME. 
153.  Time  is  the  measure  of  duration. 


60  seconds  (sec.) 
60  minutes 
24  hours    . 
7  days 
4  weeks  . 
52  weeks  . 
365  days 
100  years    . 


TABLE. 

equal  1  minute,  min 

1  hour,  h. 

1  day,  da. 

1  week,  wk. 

1  month,  mo. 

1  year,  yr. 
1  common  year,     yr. 

1  century,  cen. 


How  many 

1.  Minutes  in  1  day  ? 

2.  Seconds  in  1  day  ? 

3.  Hours  in  1  year? 

4.  Minutes  in  1  year? 

5.  Hours  in  56780  seconds  ? 

6.  Days  in  600000  seconds  ? 


Ans.  1440. 

Ans.  86400. 

Ans.  8760. 

Ans.  525600. 

Ans.  15K  etc. 

Ans. 


DENOMINATE    NUMBERS. 


139 


CIRCULAR  MEASURE. 

154.   Circular  Measure  is  used  to  measure  angles 
and  directions,  latitude  and  longitude,  etc. 

1.  A  Circle  is  a  figure  bounded  by  a  curve 
line,  every  point  of  which  is  equally  distant 
from  a  point  within,  called  the  centre. 

2.  The  Circumference  is  the  bounding 
line  ;  any  part  of  the  circumference,  as  BC,  is 
an  arc ;  AB  is  the  diameter,  and  OC  the  radius. 

3.  For  the  purpose  of  measuring  angles,  the  circumference  is 
divided  into  3G0  equal  parts,  called  degrees;  each  degree  into  60 
equal  parts,  called  minutes;  each  minute  into  60  equal  parts,  called 
seconds. 

4.  Any  angle  at  the  centre,  as  COB,  is  measured  by  the  arc 
BC  included  between  its  sides.  A  right  angle  is  measured  by 
90  degrees ;  half  a  right  angle,  by  45  degrees,  etc. 


TABLE. 


60  seconds  (")     .  . 

60  minutes      .     .  . 

30  degrees      .     .  . 

12  signs,  or  360°,  . 

How  many 

1.  Seconds  in  24' 32"? 

2.  Seconds  in  23°  24'  15"? 
Minutes  in  1472"? 


<( 


a 


u 


r 
o 


o 
O. 


equal  1  minute, 

1  degree, .     . 

1  sign  .     .     .     .     S. 

1  circumference,    C. 


Ans.  1472". 
Ans.  84255". 
Ans.  24'  32". 
Ans.  23°  24'  15". 


4.  Degrees  in  84255"  ? 

5.  What  is  the  difference  between  the  number  of 
minutes  in  a  day  and  the  number  of  minutes  in  a  cir. 
en  inference  ? 

6.  "What  is  the  difference  between  the  number  of 
seconds  in  a  day  and  the  number  of  seconds  in  a  cir- 
cumference ? 

7.  If  you  study  6  hours  a  day,  for  5  days  in  a  week, 
how  many  minutes  will  you  study  in  a  week  ? 


liO 


DENOMINATE    NUMBERS. 


MISCELLANEOUS  TABLES. 
155.     Counting. 


12  units 
12  dozen 
12  gross 
20  units 

< 

24  sheets 

20  quires 

2  reams 

5  reams 


equal  1  dozen. 
1  gross. 
1  great  gross. 
1  score. 


a 

i 
it 


156.     Paper. 

.     .     .     equal  1  quire. 
1  ream. 
1  bundle. 
1  bale. 


u 
a 


<< 


157.  Weight,  Capacity,  Length,  etc. 

56  pounds  of  rye  or  corn  equal  L  bushel. 

60  pounds  of  wheat  or  clover  seed 

60  pounds  of  beans  or  potatoes 
100  pounds  of  fish 

196  pounds  of  flour  " 

220  pounds  of  shad  or  salmon  " 

200  pounds  of  other  fish  " 

200  pounds  of  beef  or  pork  " 

14  pounds  (by  English  law)  " 

8  bushels  of  wheat  " 
4  inches  " 

9  inches  " 
22  inches  (in  Scripture)                         " 

A  knot  or  nautical  mile  is  6086.7  feet. 

A  surveyor's  chain  of  100  links  is  4  rods  long. 

1.  How  many  units  in  a  gross? 

2.  How  many  pins  in  a  great  gross? 

3.  How  many  sheets  in  a  ream  ?     In  a  bundle? 

4.  How  many  sheets  in  a  bale  ?     In  12  bales  ? 

5.  How  many  bushels  of  rye  will  weigh  as  much  aa 
14  bushels  of  wheat  ?  Ans.  15bu. 

6.  How  many  bushels  of  beans  will  weigh  as  much  as 
30  bushels  of  corn  ?  Ans.  28bu. 

7.  If  Dr.  "Windship  lifts  3000  pounds,  how  many  bar« 
rels  of  beef  can  he  lift  ?  Ans.  15  barrels. 


1  bushel. 

1  bushel. 

1  quintal. 

1  barrel. 

1  barrel. 

1  barrel. 

1  barrel. 

1  stone. 

1  quarter. 

1  hand. 

1  span. 

1  cubit. 


DENOMINATE    NUMBERS.  14] 

MISCELLANEOUS   PROBLEMS. 

1.  Reduce  907  pence  to  pounds. 

2.  Reduce  184U  pence  to  pounds. 

3.  Reduce  24S0  farthings  to  shillings. 

4.  How  many  pounds  in  8000  grains  Troy  ? 

5.  How  many  pounds  in  10000  ounces  Avoirdupois? 

6.  Reduce  £12  9s.  Gd.  to  pence. 

7.  Reduce  81b.  7oz.  lopwt.  to  pennyweights. 

8.  Reduce  121b.  14oz.  15dr.  to  drams. 

9.  How  many  seconds  in  24  hours,  or  one  day  ? 

10.  How  many  pounds  in  16cwt.  3qr.  131b.  ? 

11.  How  many  tons  in  9876  pounds? 

12.  How  many  tons  in  165762  ounces? 

13.  Change  63  29  12gr.  to  grains. 

14.  Reduce  9^  4  3  19  lOgr.  to  grains. 

15.  How  many  pounds  in  5876^  ? 

16.  How  many  pounds  in  765429  ? 

17.  Reduce  3m.  7fur.  4rd.  2yd.  to  yards. 

18.  Reduce  47692  feet  to  miles. 

19.  Reduce  1234560  inches  to  miles. 

20.  Adam  died  at  the  age  of  930  years ;  how  many 
seconds  was  this? 

21.  Methuselah  died  at  the  age  of  969  years;  how 
many  seconds  old  was  this  ? 

22.  If  the  pulse  beat  75  times  a  minute,  how  often 
does  it  beat  in  a  day?  Ans.  108000  times. 

23.  How  long  will  it  take  to  count  a  million,  at  the 
rate  of  a  hundred  a  minute,  working  12hrs.  a  day? 

Ans.  13d.  lOh.  40m. 

24.  If  the  distance  around  the  earth  is  25000  miles, 
how  long  will  it  take  to  walk  the  distance,  walking  4 
miles  an  hour?  Ans.  260d.  lOh. 

25.  If  £1  equals  84.84,  what  is  the  value  of  £5  in 
United  States  Money? 


142  DENOMINATE    NUMBERS. 

26.  If  £1  equals  $4.84,  required  the  value  of  £7  15s 
in  the  money  of  the  United  States.  Ans.  $37.51. 

27.  If  £2  equals  $9.68,  what  is  the  value  of  $37.51  in 
English  Money?  Ans   £7  15s. 

28.  If  12  of  Henry's  peaches  fill  a  quart  measure,  how 
many  will  there  be  in  a  bushel?  Ans.  384. 

29.  How  many  square  rods  in  a  rectangular  field  32 
rods  long  and  12  rods  wide?  Ans.  384sq.  rd. 

30.  How  many  square  feet  in  a  board  18  feet  long  and 
2|  feet  wide?  Ans.  45sq.  ft. 

31.  How  many  cubic  feet  in  a  block  of  stone  6  feet 
lonjj,  3  feet  wide,  and  2  feet  thick?  Ans.  36cu.  ft. 

32.  Required  the  value  of  a  rectangular  lot  36  rods 
long  and  20  rods  wide,  at  $3  a  square  rod.     Ans.  $2160., 

33.  How  many  cords  in  a  pile  of  wood  48  feet  long, 
4  feet  wide,  and  4  feet  high  ?  Ans.  6  cords. 

34.  How  many  cords  in  a  pile  of  wood  16  feet  long, 
8  feet  high,  and  4  feet  wide?  Ans.  4  cords. 

35.  What  must  I  pay  for  a  pile  of  wood  24  feet  long, 
12  feet  high,  and  4  feet  wide,  at  $1.50  a  cord? 

Ans.  $13.50. 

36.  How  much  time  is  wasted  by  taking  an  hour's 
nap  each  afternoon,  for  24  years  of  365  days  each  ? 

Ans.  1  year. 

ADDITION  OF  DENOMINATE  NUMBERS. 
158,   Addition  of  Denominate  Numbers  is  the  pro- 
cess of  finding  the  sum  of  two  or  more  denominate 
numbers  of  the  same  kind  of  quantity. 

1.  Find  the  sum  of  £8  7s.  5d.;  £9  8s.  6d. ;  £7  14s.  9d 
Solution. — We    write    the    numbers   so   that         operation. 
units  of  the  same  kind  shall  stand  in  the  same  £     s.    d. 

column,  and  begin  at  the  right  to  add.     9d  plus  8     7     5 

6d.  plus  5d.  equal  20d.,  which,  by  reduction,  we  9     8     6 

find  equal  Is.  and  8d.     We  write  the  8d.  under  7  14     9 

the  pence  column,  and  reserve  the  Is.  to  add  to         25  10     8 
the  column  of  shillings.     Is.  plus  14s.  plus  8s. 


DENOMINATE    NUMBERS.  143 

plus  7s.  equal  30s.,  which,  by  reduction,  we  find  equal  £1  and  10s. 
We  write  the  10s.  in  shillings  column,  and  add  the  £1  to  the  column 
of  pounds,  etc.     Hence  the  following 

Rule. — 1.  Write  the  numbers  so  that  units  of  the  same 
name  stand  in  the  same  column,  and  commence  at  the  right 
to  add. 

2.  Add  as  in  simple  numbers,  reduce  by  division  the  sum 
of  each  column  to  the  next  higher  denomination,  write  the 
remainder  under  the  column  added,  and  add  the  quotient  to 
the  next  column. 

3.  Proceed  in  the  same  manner  icith  all  the  columns  to 
the  last,  under  which  write  the  entire  sum. 

Proof. — The  same  as  in  simple  numbers. 


(2.) 

(3.) 

(4-) 

£   s.   d. 

£ 

s.   d. 

£ 

s.   d. 

24  12  6 

25 

16  8 

123 

14  6 

25  13  9 

17 

13  9 

137 

18  10 

17  18  10 

14 

17  11 

246 

19  11 

(5.) 

(6.) 

(70 

lb.  oz.  pwt. 

lb. 

oz.  pwt. 

lb. 

oz.  pwt. 

17  9  16 

18 

9  16 

92 

7  12 

25  6  12 

36 

8  21 

71 

3  17 

72  11  13 

29 

7  23 

28 

9  10 

57  10  19 

42 

11  17 

36 

11  18 

(8) 

(9.) 

(10.) 

cwt.  qr.  lb.   oz. 

qr. 

lb.   oz. 

dr. 

rd. 

yd.  ft.  in 

20  3  12  11 

12 

16  12 

11 

17 

4  2  6 

16  2  16  12 

13 

23   9 

10 

21 

2  17 

17  0  22  20 

14 

24  14 

15 

23 

3  0  8 

19  1  18  19 

15 

16  15 

8 

25 

5  2  9 

144  DENOMINATE    NUMBERS. 

(11.)  (12.)  (13.) 


ft). 

5 

9 

gr- 

gal. 

qt. 

pt. 

L. 

mi. 

fur. 

rd. 

28 

11 

7 

2 

16 

36 

2 

1 

16 

2 

7 

30 

19 

9 

5 

1 

23 

42 

1 

1 

14 

2 

7 

32 

27 

8 

3 

2 

17 

25 

3 

0 

28 

1 

6 

28 

24 

7 

2 

1 

18 

28 

3 

1 

34 

0 

5 

37 

SUBTRACTION  OF  DENOMINATE  NUMBERS. 

159.  Subtraction  of  Denominate  Numbers  is  the 

process  of  finding  the  difference  between  two  compound 
numbers  of  the  same  kind  of  measure. 

1.  From  lOoz.  12pwt.  20gr.  take  7oz.  15pwt.  16gr. 

Solution. — We  write  the  subtrahend  under         operation. 
the  minuend,  writing  units  of  the  same  name  in         oz.  pwt.  gr. 
the  same  column,  and  commence  at  the  lowest         19     12     20 
denomination    to    subtract.       16gr.    subtracted  7     15     16 

from  20gr.  leave    4grs.,  which  we  write  under  2     17       4 

the  grains.  15pwt.  from  12pwt.  we  cannot 
take;  we  will  therefore  take  loz.  from  the  lOoz.,  leaving  9oz. ;  'ioz. 
equal  20pwt.,  which  added  to  12pwt.  equal  32pwt. ;  15pwt.  sub- 
tracted from  32pwt.  equal  17pwt.,  which  we  write  under  pwts.  7oz. 
from  9oz.  (or  since  it  will  give  the  same  result,  we  may  add  loz.  to 
7oz.  and  say,  8oz.  from  lOoz.)  leave    2oz.     Hence  the  following 

Rule. — 1.  Write  the  subtrahend  under  the  minuend,  with 
units  of  the  same  denomination  in  the  same  column. 

2.  Commence  at  the  lowest  denomination,  and  subtract 
each  number  in.  the  subtrahend  from  the  corresponding 
number  in  the  minuend. 

3.  If  the  number  in  the  subtrahend  exceeds  the  number  in 
the  minuend,  add  to  the  latter  as  many  units  of  that  deno- 
mination as  make  one  of  the  next  higher,  and  then  subtract ; 
add  also  one  to  the  next  number  in  the  subtrahend  before 
subtracting. 

4.  Proceed  in  the  same  manner  with  each  denomination 
to  the  Last. 

Proof. — The  same  as  in  simple  numbers. 


DENOMINATE    NUMBERS.  145 


(2-) 
£    s.   d.  far. 

(3.) 
£    s. 

d.  far. 

lb. 

(4.) 
oz.  pwt.  gr 

143  11  10  2 

930  17 

7  3 

16 

10  16  18 

115  14   6  3 

246  19 

8  1 

13 

11  17  15 

27  17   3  3 

2 

10  19   3 

(5.) 
lb.  oz.  pwt.  gr. 

125  8  14  20 

(6.) 
cwt.  qr.  lb. 
112  3  17 

oz. 
12 

T. 
236 

cwt.  qr.  lb.  oz 
13  2  18  12 

96  9  10  23 

37  1  10 

13 

127 

11  4  22  10 

(8.) 
hhd.  gal.  qt.  pt. 

128  27  0  1 

(9.) 
yr.  mo.  wk 
216  10  2 

.  da.  h. 

5  16 

(10.) 

sq.yd.  sq.ft.  sq.in 

226   20   120 

106  30  2  1 

123  10  3 

2  20 

S. 

L34  25  130 

(11.) 
A.   II.  P. 

(12.) 
L.  mi.  fur. 

rd. 

(13.) 

o      /     // 

426  1  30 

16  2  7 

30 

25 

20  30  40 

207  3  35 

14  2  7 

32 

20 

30  40  50 

14.  A  farmer  had  200bu.  of  wheat,  and  sold  28bu.2pk. 
5qt.  lpt.  to  one  man,  and  as  much  more  to  another ;  how 
much  remained  ?  Ans.  142bu.  2pk.  5qt. 

15.  A  miner  having  1121b.  of  gold  sent  his  mother 
171b.  lOoz.  15pwt.  20gr.  and  31b.  16pwt.  less  to  his 
father ;  how  much  did  he  retain  ? 

Ans.  791b.  3oz.  4pwt.  8gr. 

16.  Subtract  16dol.  57cts.  5*  mills  from  $25  20cts.  ~\ 
mills,  and  add  2  eagles  and  25  £  dimes  to  the  result. 

Ans.  31dol.  20cts.  6|  mills. 
IT.  Add  961b.  9oz.  lOpwt.  23gr.  to  1251b.  8oz.  14pwt 
20grs.   and   subtract  the  sum  from  the  sum  of  1021b. 
iloz.  16pwt.  and  2561b.  9oz.  19pwt. 

13 


146  DENOMINATE    NUMBERS. 

MULTIPLICATION  OF  DENOMINATE  NUMBERS 

16©.  Multiplication  of  Denominate  Numbers  is  the 

process  of  multiplying  a  denominate  number  by  an  ab- 
stract number. 

1.  Multiply  £12  lis.  7d.  by  8. 

Solution. — 8  times  7d.  are  56d.,  which  by  operation. 

reduction  we  find  is  equal  to  4s.  and  8d.     We  £      s.     d. 

write  the  8  pence  under  the  pence,  and  re-  -            12     11     7 

serve  the  4s.  to  add  to  the  next  product.     8  8 

times  lis.  are  88s.,  which  added  to  the  4s.  ^(X)     12     8 
equal   92s.,  which  we  find  by  reduction    equal 

£4  and  12s.     8  times  £12  are  £96,  which  added  to  £4  equal    £100, 
Hence  the  following 

Rule.  —  Write  the  multiplier  under  the  lowest  denomina- 
tion of  the  multiplicand,  multiply  as  in  simple  number^ 
reducing  as  in  addition. 

Proof. — The  same  as  in  simple  numbers. 

EXAMPLES   FOR  PRACTICE. 

(2.)  (3.)  (4.) 

cwt.  qr.    lb.     oz.  lb.    oz.   pwt.  gr.  M.      da.      h.  min.  pec. 

18     3     21     9  16     8     15  17  50     10     20  30     40 

5  3  7 


(5.)  (6.)  (7.) 

£        s.        d.    far.  hhd.   gal.    qt.  pt.  lb      3      3      J}  gr. 

13     12     9     2  21     35     3  1  12     8     7     2  20 

8  9  11 


8.  Multiply  12L.  2mi.  5fur.  32rd.  by  5,  by  6,  by  7,  by  8 

9.  Multiply  23ch.  18bu.  2pk.  7qt.  lpt.  by  4,  by  5,  by 
9,  by  10. 

10.  A  farmer  sold  5  loads  of  hay,  each  containing 
15cwt.  3qr.  151b. ;  how  much  did  he  sell  ? 

Ans.  79cwt.  2qr. 


DENOMINATE    NUMBERS.  147 

11.  Multiply  13yr.  lOmo.  3wk.  5da.  by  5,  and  that 
product  by  3.  Ans.  208yr.  "mo.  3wk.  5da. 

12.  If  a  man  walk  17mi.  7  fur.  20rd.  in  each  of  21 
days,  how  far  will  he  walk  in  all? 

Ans.  3T6mi.  5fur.  20rd. 

13.  If  a  farmer  raise  60bu.  3pk.  6qt.  lpt.  of  grain  on 
one  acre,  how  much  can  he  raise  at  the  same  rate  on 
48  acres?  *  Ans.  2925bu.  3pk. 

14.  A  owned  1000A.  of  land ;  he  sold  B  96A.  3K. 
30P.,  and  4  times  as  much  to  C  ;  how  much  remained  ? 

Ans.  515A.  1H.  10P. 


DIVISION  OF  DENOMINATE  NUMBERS. 


161.  Division  of  Denominate  Numbers  is  the  pro- 
;ss  of  dividing  when  one  or  bot 
number.     There  are  two  cases. 


cess  of  dividing  when  one  or  both  terms  is  a  denominate 


Case  I. 

162.  To  divide  a  denominate  number  into  equal 
parts. 

1.  Divide  £103  7s.  6d.  into  5  equal  parts;  that  is,  take 
l  of  it. 

Solution. — \  of  £103  is  £20  and  £3  re-  operation. 

maining.     £3  equal    GOs.,  which  added  to  7s.  £       s.    d. 

equal    67s. ;  i  of  67s.  is  13s.  and  2s.  remain-  5)103     7     6 

ing.     2s.    equal     24d.,    which    added    to   6d.  20  13     6 
equal     30d. ;    \  of  30d.   is   6d.     Hence   the 
following 


*o 


Rule. — 1.  Begin  at  the  highest  denomination,  and  divide 
as  in  simple  numbers. 

2.  If  there  is  a  remainder,  reduce  it  to  the  next  loioer  de-. 
nomination,  add  to  it  the  number  of  that  denomination 
and  divide  as  before,  and  thus  continue  to  the  last. 


148  DENOMINATE    NUMBERS. 


EXAMPLES 

FOR 

PRACTICE. 

(2.) 

(3.) 

(4.) 

£ 

s.      d. 

lb. 

oz. 

pwt. 

gr. 

T. 

cwt.   qr. 

A. 

4)61 

18     4 

9     7 

6)76 

10 

14 

12 

7)112 

!      16     2 

16 

15 

16 

2     1 

13 

(5.) 

(6.) 

cwt. 

qrs.     lb. 

oz. 

dr. 

hhd.     gal. 

qt.       pt. 

g1- 

8)125 

3     19 
(7.) 

12 

8 

9)1' 

08     42 

2       1 

2 

(8.) 

min. 

fur.    rd. 

yd. 

ft. 

A. 

R.      P.  sq.yd. 

11)120 

7     33 

3 

2 

5)112 

3     24 

(10.) 

24 

(9.). 

• 

bu. 

pk.     qt 

.    Pt. 

lb. 

oz.     pwt. 

gr- 

9)1137 

3      4 

1 

8)37 

10     17 

16 

11.  A  miner  divides  371b.  lOoz.  17pwt.  16gr.  of  gold 
among  8  sisters ;  how  much  does  each  receive  ? 

Ans.  41b.  8oz.  17pwt.  5gr. 

12.  A  man  walked  376mi.  6fur.  36rd.  in  22  days;  what 
was  the  average  distance  each  day  ? 

Ans.  17 mi.  lfur.  lT77rd. 

13.  If  26  casks  contain  21hhd.  llgal.  2qt.  lpL,  what 
is  the  capacity  of  each  cask?      Ans.  51gal.  lqt.  Upt. 

Case  II. 

163.  To  divide  a  denominate  number  by  a  simi- 
lar denominate  number. 

1.  Divide  £26  6s.  2d.  by  £4  15s.  8d. 

Solution.— £26  6s.  2d.  we  find  by  operation. 

reduction  equal   6314  pence ;  £4  15s.  £26    6s.    2d.  =  6314d. 

8d.  equals  1148  pence;  and  dividing  £415s.    8d.  =  1148d. 

6314d.  by  1148d.  we  obtain  a  quotient  1148)6314f51    Ans. 

of  5£.     From  this  solution  we  have  6314 
the  following 

Bule.— Reduce  both  dividend  and  divisor  to  the  lowest 


DENOMINATE    NUMBERS.  149 

denomination  mentioned  in  either,  and  then  divide  as  in 
simple  numbers. 

Remark. — The  division  may  also  be  made  before  the  reduction  to 
lower  denomination ;  and  this  will  be  shorter  when  there  is  no  re- 
mainder. 

2.  Divide  £48  7s.  4d.  by  £6  lid.  Ans.  8. 

3.  Divide  69bu.  3pk.  6qt.  by  6bu.  3pk.  6qt. 

Ans.  10^. 

4.  Divide  80bu.  2pk.  4qt.  by  13bu.  3pk.  5qt. 

.ci.ns.  Ojt-jj. 

5.  Divide  6971b.  7oz.  5dr.  by  601b.  lOoz.  6dr. 

Ans.  11A. 

6.  A  man  travelled  3mi.  6fur.  36rd.  4yd.  in  one  hour; 
in  what  time  will  he  travel  247mi.  2fur.  30rd.  3yd.  ? 

Ans.  64  hrs. 

7.  A  drove  of  cattle  ate  6T.  15cwt.  3qr.  121b.  of  hay 
in  a  week ;  how  long  will  33T.  19cwt.  lqr.  101b.  last 
them  ?  Ans.  5  weeks. 

PROBLEMS   IN   TIME. 

1.  'Washington  was  born  Feb.  22d,  1732,  and  died 
Dec.  14th,  1799;   what  was  his  age? 

Solution. — We  write  the  number  of  the 
year,  month,  and  day  of  both  periods,  and 
subtract  the  one  from  the  other,  as  is  shown 
in  the  margin. 

67       9     22 

2.  John  Adams  was  born  the  19th  of  October,  1735, 
and  died  the  4th  of  July,  1826 ;  required  his  age. 

3.  Thomas  Jefferson  was  born  xVpril  2d,  1743,  and 
died  July  4th,  1826;  what  was  his  age? 

4.  James  Madison  was  born  March  16th,  1751.  and 
died  June  28th,  1836 ;  required  his  age. 

5.  James  Monroe  was  born  April  28th,  1758,  and  died 
July  4th,  1831 ;  required  his  age. 

13* 


OPERATION. 

yr- 

mo. 

da. 

1799 

12 

14 

1732 

2 

22 

150  DENOMINATE    NUMBERS. 

6.  John  Quincy  Adams  was  born  July  11th,  1767,  and 
died  Feb.  23d,  1848 ;  what  was  his  age? 

7.  Andrew  Jackson  was  born  March  15th,  1767.  and 
died  June  8th,  1845;  required  his  age. 

8.  Martin  Van  Buren  was  born  Dec.  5th,  1782,  and 
died  July  24th,  1862 ;  required  his  age. 

9.  William  Henry  Harrison  was  born  Feb.  9th,  1773, 
and  died  April  4th,  1841 ;  required  his  age. 

10.  James  K.  Polk  was  born  Nov.  2d,  1795,  and  died 
June  15th,  1849 ;  required  his  age. 

11.  General  Zachary  Taylor  was  born  Nov.  24th,  1784, 
and  died  July  9th,  1850 ;  required  his  age. 

12.  How  Ions  has  a  note  to  run  which  is  dated  Dec. 
30th,  1862,  and  made  payable  Jan.  16th,  1864? 

Ans.  lyr.  16da. 

13.  The  Revolution  was  commenced  the  19th  of  April, 
1775,  and  terminated  January  20th,  1783  ;  how  long  did 
it  continue?  Ans.  7yr.  9mo.  Ida. 


PROBLEMS  IN  LATITUDE  AND  LONGITUDE. 

1.  The  latitude  of  Boston  is  42°  21'  23"  north ;  that 
of  Charleston,  32°  46'  33"  north ;  what  is  the  difference 
of  latitude  ? 

2.  The  latitude  of  New  York  is  40°  24'  40"  K;  that 
of  New  Orleans,  29°  57'  30"  N. ;  what  is  the  difference 
of  latitude  ? 

3.  The  latitude  of  Philadelphia  is  39°  56'  39" ;  that 
of  Savannah  is  32°  4'  56";  what  is  the  difference  of 
latitude  ? 

4.  The  latitude  of  Baltimore  is  39°  17'  23";  that  of 
St.  Louis  is  38°  37'  28";  what  is  the  difference  of 
latitude  ? 

5.  The  latitude  of  the  Cape  of  Good  Hope  is  30°  55' 
15"  S.;  that  of  Cape  Horn,  55°  58'  30";  what  is  the 
difference  of  latitude  ? 


INTRODUCTION 


TO    THE 


METRICAL   SYSTEM   OF   WEIGHTS    AND   MEASURES. 


The  old  system  of  weights  and  measures  in  our  country  is  irregular^ 
difficult  to  learn,  and  inconvenient  to  apply.  The  same  is  true  with 
She  old  systems  of  all  nations.  Originating  by  chance,  rather  than 
science,  they  lacked  the  simplicity  of  law,  and  were,  therefore,  irregu- 
lar and  chaotic. 

In  1795,  France  adopted  a  system  of  weights  and  measures  called 
the  Metric  System,  based  upon  the  decimal  method  of  notation ;  all 
the  divisions  and  multiples  being  by  10.  It  was  regarded  as  so 
great  an  improvement  upon  the  old  methods  that  it  has  since  been 
adopted  by  Spain,  Belgium,  and  Portugal,  to  the  exclusion  of  all 
other  weights  and  measures,  and  is  in  partial  use  in  Holland,  Italy, 
Germany,  and  Austria,  and  also  in  many  parts  of  Spanish  America. 

In  1864,  the  British  Parliament  passed  an  act  permitting  its  use 
throughout  the  empire  whenever  parties  should  agree  to  use  it.  In 
18G6,  Congress  authorized  its  use  in  the  United  States,  and  provided 
for  its  introduction  into  the  post-offices  for  the  weighing  of  letters 
and  papers. 

To  facilitate  its  adoption,  a  convenient  standard  of  comparison  was 
furnished,  by  making  the  new  five-cent  piece  five  grams  in  weight 
and  one  fiftieth  of  a  meter,  or  two  centimeters,  in  diameter.  This 
system  will,  without  doubt,  in  a  few  years  be  in  general  use  in  this 
country. 

The  advantages  of  the  Metric  System  are  numerous  and  important. 

1.  It  is  easily  learned ;  a  school-boy  can  learn  it  in  a  single  after- 
noon. 

2.  It  is  easily  applied ,  all  the  operations  being  the  same  as  in 
simple  numbers. 

3.  It  does  away  with  addition,  subtraction,  multiplication,  division, 
and  reduction  of  compound  numbers  and  fractions. 

4.  It   will  facilitate   commerce,   giving   the   nations    a   universal 

system  of  weights  and  measures. 

151 


152 


THE    METRIC    SYSTEM. 


THE   METEIC   SYSTEM. 

164.  The  Metric  System  of  Weights  and  Measures  is 
based  upon  the  decimal  system  of  notation. 

k«3.  In  this  system  we  first  establish  the  unit  of 
each  measure,  and  then  multiply  and  divide  it  by  10. 

166.  Names. — We  first  name  the  unit  of  any  measure, 
and  then  derive  the  other  denominations  by  prefixing 
words  to  the  unit  flame. 

167.  The  higher  denominations  are  expressed  by  pre- 
fixing to  the  name  of  the  unit, 


Deka, 

10 


Hecto, 

100 


Kilo, 

1000 


Myria. 

10,000 


168.  The  lower  denominations  are  expressed  by  pre- 
fixing to  the  name  of  the  unit, 


I>eci, 

l^ 

10 


Centi, 

l 

loo 


Milli. 

l 


1000 


169.   Units. — The  following  are  the  different  units, 
with  their  English  pronunciation : — 


Measure. 

Unit. 

Pronunciation. 

Measure. 

Unit. 

Pronunciatio 

Length, 

Meter, 

(meter.) 

Capacity, 

Liter, 

(leeter.) 

Surface, 

Are, 

(air.) 

Weight, 

Gram, 

(gram.) 

Volume, 

Stcre, 

(stair.) 

Value, 

Dollar* 

MEASURE    OF   LENGTH. 
170.  The  Meter  is  the  unit  of  length.      It  is  the  ten- 
millionth  part  of  the  distance  from  the  equator  to  the 
poles,  and  equals  39.37  inches,  or  3.28  feet. 


TABLE. 

10  millimeters  (m.m.)  equal  1  centimeter, 


10  centimeters 
10  decimeters 
10  meters 

10  dekameters 
10  hectometers 
10  kilometers 


u 


a 


a 


a 


a 


n 


1  decimeter, 
1  meter, 

1  dekanieter, 
1  hectometer, 
1  kilometer, 


cm. 

d.m. 

M. 

D.M. 

H.M. 

K.M. 


1  myriameter,    M.M. 


THE    METRIC    SYSTEM.  153 

Notes. — 1.  The  meter  is  very  nearly  3  feet,  3  inches,  and  3  eighths  of 
an  inch  in  length,  which  may  be  easily  remembered  as  the  rule  of 
three  threes. 

2.  Cloth,  etc.  are  measured  by  the  meter;  very  small  distances,  by 
the  millimeter;  great  distances,  by  the  kilometer. 

3.  The  new  5-cent  piece  is  -fa  of  a  meter  in  diameter:  hence  its 
diameter  is  ^  of  a  decimeter,  or  2  centimeters. 

4.  A  decimeter  is  about  4  inches ;  a  kilometer,  about  200  rods,  or 
|  of  a  mile  ;  a  millimeter,  about  ^  of  an  inch.  The  inch  is  about  2i 
centimeters ;  the  foot,  3  decimeters ;  the  rod,  5  meters ;  the  mile, 
1600  meters,  or  16  hectometers. 

QUESTIONS. 

1.  How  many  centimeters  in  a  meter? 

2.  How  many  millimeters  in  a  meter? 

3.  How  many  decimeters  in  a  dekameter? 

4.  How  many  meters  in  a  hectometer  ? 

5.  How  many  meters  in  a  kilometer  ? 

MEASURES   OF   SURFACE. 

171.  The  Are  is  the  unit  of  surface  used  to  measure 
land.  The  are  is  a  square  dekameter.  It  equals  119.6  sq. 
yd.,  or  0.0247  acre. 


TABLE. 

10  milliares  (m. 

,a.)  equal  1  centiare, 

c.a. 

10  centiares 

u 

1  deciare, 

d.a. 

10  declares 

a 

1  are, 

A. 

10  ares 

tt 

1  dekare, 

D.A. 

10  dekares 

u 

1  hectare, 

H.A. 

10  hectares 

it 

1  kilare, 

K.A. 

10  kilares 

u 

1  myriare, 

M.A. 

Notes. — 1.  The  are,  centiare,  and  hectare  are  the  denominations 
principally  used,  as  these  are  exact  squares.  The  centiare  is  a 
square  whose  side  is  1  meter ;  the  hectare  is  a  square  whose  side  is 
100  meters. 

The  are  =  100  square  meters.      The  centiare  =  1  square  meter. 
The  hectare  =  10,000  square  meters. 

2.  The  deciare  is  not  a  square,  it  is  merely  the  tenth  of  an  are; 
the  dekare  is  not  a  square,  it  is  merely  10  ares. 


154  THE    METRIC    SYSTEM. 

3.  A  hectare  equals  very  nearly  21  acres ;  a  centiare  equals  nearly 
1  \  sq.  yd.     An  acre  is  very  nearly  40  ares. 

MEASURES  OF  OTHER  SURFACES. 

172.  All  surfaces  besides  land  are  measured  by  the 
square  meter,  square  decimeter,  etc.  The  measures  are 
shown  by  the  following  table : — 


TABLE. 

2 

2 

■ 

2 


100  sq.  millimeters  (m.m.2)  =  l  sq.  centimeter,  cm. 
100  sq.  centimeters  =1  sq.  decimeter,    d.m. 

100  sq.  decimeters  =1  sq.  meter,  M. 

Note. — The  measures  higher  than  these  are  not  generally  usod. 

QUESTIONS. 

1.  How  many  centiares  in  an  are? 

2.  How  many  ares  in  a  hectare  ? 

3.  How  many  square  meters  in  an  are? 

4.  How  many  square  decimeters  in  an  are? 

5.  How  many  ares  in  640  square  meters  ? 

MEASURES   OF   VOLUME. 
173.  The  Stere  is  the  unit  of  volume.     It  is  a  cubic 
meter,  and  equals  35.3166  cubic  feet,  or  1.308  cu.  yd. 

TABLE. 

10  millisteres  (m.S.)  equal  1  centistere,  c.S. 

10  centisteres  "  1  decistere,  d.s. 

10  decisteres  "  1  stere,  S. 

10  steres  "  1  dekastere,  D.S. 

10  dekasteres  "  1  hectostere,  H.S. 

10  hectosteres  "  1  kilostere,  K.S. 

10  kilosteres  "  1  myriastere,  M.S. 

Note.— 1.  Wood  is  measured  by  this  measure.  The  stere,  ieci- 
&tere,  and  dekastere  are  principally  used.  3.6  steres,  or  36  deci- 
steres, very  nearly  equal  the  common  cord. 


THE    METRIC    SYSTEM.  155 

MEASURES    OF    OTHER    VOLUMES. 

174,  Other  solid  bodies  are  usually  measured  by  the 
cubic  meter  and  its  divisions.  The  measures  are  shown 
by  the  following  table. 

TABLE. 

1000  cubic  millimeters(m.m.3)=l  cubic  centimeter.c.m.* 
1000  cubic  centimeters  =1  cubic  decimeter,  d.m.3 

1000 cubic  decimeters  =1  cubic  meter,         M.3 

N0TE. — The  higher  denominations  are  not  generally  used. 

QUESTIONS. 

1.  How  many  centisteres  in  a  stere  ? 

2.  How  many  decisteres  in  a  dekastere  ? 

3.  How  many  dekasteres  in  a  kilostere  ? 

4.  How  many  cubic  meters  in  a  hectostere? 

MEASURES    OF    CAPACITY. 

175.  The  Liter  is  the  unit  of  capacity.  It  equals  a 
cubic  decimeter;  that  is,  a  cubic  vessel  whose  size  is  one- 
tenth  of  a  meter. 

This  measure  is  used  for  measuring  liquids  and  dry 
substances.  The  liter  is  a  cylinder,  and  holds  2.1135 
pints  wine  measure,  or  1.816  pints  dry  measure. 

TABLE. 


10  milliliters  (m. 

,1.)  equal  1  centiliter, 

C.I. 

10  centiliters 

u 

1  deciliter, 

d.l. 

10  deciliters 

u 

1  liter. 

L. 

10  liters 

it 

1  dekaliter, 

D.L. 

10  dekaliters 

a 

1  hectoliter, 

H.L. 

10  hectoliters 

« 

1  kiloliter, 

K.L. 

10  kiloliters 

u 

1  myrialiter. 

M.L. 

Notes. — 1.  The  liter  is  principally  used  in  measuring  liquids,  and 
the  hectoliter  in  measuring  grains,  etc. 

2.  The  liter  equals  nearly  1J5  liquid  quarts,  or  T9j  of  a  dry  quart. 
or  nearly  -^  of  a  bushel  measure. 


156  THE    METRIC    SYSTEM. 

3.   The  hectoliter  is  about  2|  bushels,  or  |  of  a  barrel.     4  liters  are 
a  little  more  than  a  gallon;  35  liters,  very  nearly  a  bushel. 

QUESTIONS. 

1.  How  many  liters  in  a  hectoliter? 

2.  How  many  liters  in  a  kiloliter? 

3.  How  many  deciliters  in  a  dekaliter  ? 

4.  How  many  liters  in  a  cubic  meter?  Ans.  1000. 

5.  How  many  liters  in  a  stere  ?  Ans.  1000. 

MEASURE    OF    WEIGHT. 

176.  The  Gram  is  the  unit  of  weight.  It  is  the  weight 
of  a  cubic  centimeter  of  distilled  water  at  the  tempera- 
ture of  melting  ice.    The  gram  equals  15.432  troy  grains. 


TABLE. 

10  milligrams  (m 

.g.)  equal  1  centigram, 

e.g. 

10  centigrams 

a 

1  decigram, 

d.g. 

10  decigrams 

a 

1  gram, 

G. 

10  grams 

u 

1  dekagram, 

D.G. 

10  dekagrams 

u 

1  hectogram, 

H.G. 

10  hectograms 

a 

1  kilogram, 

K.G.,orK. 

10  kilograms 

to 

1  myriagram, 

M.G. 

Notes. — 1.  The  gram  is  used  in  weighing  letters,  in  mixing  and 
compounding  medicines,  and  in  weighing  all  very  light  articles. 
The  new  5-cent  coin  (dated  1866)  weighs  5  grams. 

2.  The  kilogram  is  the  ordinary  unit  of  weight,  and  is  generally 
abbreviated  into  kilo.  It  equals  about  2^  pounds  avoirdupois. 
Meat,  sugar,  etc.  are  bought  and  sold  by  the  kilogram. 

3.  In  weighing  heavy  articles,  two  other  weights,  the  quintal 
(100  kilograms)  and  the  tonneau  (1000  kilograms),  are  used.  The 
tonneau  is  between  our  short  ton  and  long  ton. 

4.  The  avoirdupois  ounce  is  about  28  grams;  the  pound  is  a  little 
less  than  J  a  kilo. 

QUESTIONS. 

1.  How  many  grams  in  a  kilogram? 

2.  How  many  milligrams  in  a  gram  ? 

3.  How  many  decigrams  in  *•>  kilogram? 

4    How  many  hectograms  in  a  myriagram  ? 


THE    METRIC    SYSTEM.  157 

NUMERATION   AND   NOTATION. 

177.  In  the  Metric  System  the  decimal  point  is 
placed  between  the  unit  and  its  division,  tne  whole 
quantity  being  regarded  as  an  integer  and  a  decimal. 
Thus,  3  dekagrams,  5  grams,  6  decigrams,  and  8  centi- 
grams, are  written  thus  :  35.68  grams. 

178.  The  initials  of  the  denomination  may  be  placed 
either  before  or  after  the  quantity ;  thus,  27  grams  may 
be  written  G.27,  or,  27 G. 

EXERCISES    IN    NUMERATION. 

L  Eead  M. 28.35. 

Solution. — This  is  read  28  and  35  hundredths  meters;  or  it  may 
be  read  2  dekameters,  8  meters,  3  decimeters,  and  5  centimeters. 

Read  the  following. 


2.  M. 15.37. 

3.  M.46.75. 

4.  A.57.34. 

5.  A. 75.25. 

6.  S.  134.09. 

7.  S. 325.125. 


8.  L.57.45. 

9.  L.68.25. 

10.  G.72.325. 

11.  G.416.318. 

12.  G. 207.305. 

13.  M.3056.705. 


EXERCISES    IN    NOTATION. 

1.  "Write  7  meters  and  5  centimeters. 

Solution. — We    write    the   7   meters    with   a 
decimal  point  to  the  right,  and  then,  since  there  operation. 

are   no  decimeters,   we  write  a  naught   in  the  M.7.05 

tenths  place,  and  then  write  the  5  centimeters 
in  the  place  of  centimeters. 

2.  Write  8  meters  and  6  centimeters.         Ans.  M.8.06. 

3.  Write  12  meters  and  56  decimeters      Ans.  M.  12.56. 

4.  Write  25  meters  and  8  millimeters.     Ans.  M. 25.008. 

5.  Write  13  ares,  3  declares,  5  centiares. 

Ans.  A. 13.35. 

6.  Write  24  ares,  5  centiares,  7  milliares. 

Ans.  A.24.057. 


158  THE    METRIC    SYSTEM. 

7.  Write  7  dekares,  4  declares,  5  centiares. 

Ans.  A. 70.4. 

8.  "Write  6  hectares,  8  declares,  2  milliares. 

Ans.  A. 600.802. 

9.  Write  25  steres,  6  decisteres,  5  centisteres. 

Ans.  S. 25.65. 

10.  Write  8  hectosteres,  7  steres,  4  centisteres. 

Ans.  S. 807.04. 

11.  Write  53  liters,  8  deciliters,  5  milliliters. 

Ans.  L.53.805. 

12.  Write  17  grams,  5  decigrams,  6  centigrams,  4  milli- 
grams. Ans.  G. 17. 564. 

13.  Write  42  kilograms,  8  dekagrams,  3  decigrams. 

*  Ans.  K.42.0803. 

14.  Write  27  kilograms,  9  grams,  5  decigrams. 

Ans.  K. 27.0095. 

15.  Write  8  myriagrams,  4  kilograms,  3  dekagrams, 
5  grams,  6  decigrams.  Ans.  K. 84.0356. 

REDUCTION. 

179.  Reduction  in  the  Metric  System  is  very  simple, 
since  the  numbers  are  expressed  in  the  decimal  system. 

ISO.  Since  ten  of  any  denomination  equal  one  of 
the  next  higher,  we  can  reduce  from  one  denomination 
to  another  by  simply  changing  the  position  of  the  deci- 
mal point. 

Case  I. 
181.  To  reduce  a  number  to  lower  denominations. 

1.  Reduce  25.75  dekameters  to  meters. 

Solution. — Since    10   meters    equal   1    deka- 

metcr,    ten    times   the   number    of   dekameters  oteratiox. 

equal  the  number  of  meters;    ten  times  23.75  D.M.25.75  = 

equal  257.5,  which  changes  the  decimal  point  one  M. 257.5 
place  to  the  right.     Hence  the  following  rule: 

.Rule. — Remove  the  decimal  point  as  many  -places  to  the 


THE    METRIC    SYSTEM.  159 

right  as  there  are  tenths  required  to  make  the  lower  denomi- 
nations. 

Reduce 

2.  47.125  hectometers  to  meters.  Ana.  M. 4712. 5. 

3.  35.25  dekagrams  to  grams.  Ans.  G.352.5. 

4.  46.75  dekaliters  to  liters.  Ans.  L.467.5. 

5.  7.0375  kilograms  to  grams.  Ans.  G. 7037. 5. 

6.  9.5063  hectares  to  ares.  Ans.  A. 950. 63. 

7.  5.6304  hectometers  to  meters.  Ans.  M. 563.04. 

8.  53.025  steres  to  centisteres.  Ans.  C.S. 5302.5. 

9.  365.24  square  meters  to  square  decimeters. 

Ans.  36524d.m2. 

10.  432.15  square  dekameters  to  square  meters. 

Ans.  M2. 43215. 

11.  356.25  ares  to  square  meters.  Ans.  M\  35625. 

12.  56  cubic  meters  to  cubic  decimeters. 

Ans.  56000c. d3. 

Case  II. 

182.  To  reduce  a  number  to  Mglaer  denomi- 
nations. 

10  Reduce  257.5  meters  to  dekameters. 

Solution. — Since    1    dekameter    equals    10 
meters,  1  tenth  of  the  number  of  meters  equals  operation. 

the  nuinber  of  dekameters;   1  tenth  of  257.5  is         M.257.5  = 
25.75,    which    changes   the    decimal   point    one      D.M.25.75 
place  to  ine  left.     Hence  we  have  the  following 
rule : 

Rule.—  Remove  the  decimal 'point  as  many  places  toward 
the  left  as  there  are  tens  required  to  make  the  higher  denomi- 
nation. 

Reduce' 

2.  4712.5  meters  to  hectometers.  Ans.  47.125H.M. 

3.  352.5  grams  to  dekagrams.  Ans.  35.25D.G. 

4.  467.5  liters  to  dekaliters.  Ans.  4('>7">D.L. 

5.  7037.5  grams  to  kilograms.  Ans.  7.0375K.G. 


IGO  THE    METRIC    SYSTEM. 

6.  950.63  ares  to  centiares.  Ans.  9. 5063c. a. 

7.  563.04  meters  to  centimeters.  Ans.  5.6304c. m. 

8.  5302.5  steres  to  centisteres.  Ans.  53.025c. s. 

9.  36524  d.m2.  to  square  meters.  Ans.  365.24  M2. 

10.  43215  M\  to  square  dekameters.  Ans.432.15D.IVl2. 

11.  35625  M\  to  ares.  Ans.  356.25  ares. 

12.  56000  c.d3.  to  cubic  meters. 

Ans.  56  cubic  meters. 


ADDITION   IN   THE   METRIC   SYSTEM. 
183.  The  Metric  System  being  founded  upon  the  deci- 
mal system,  addition  is  performed  the  same  as  in  simple 
numbers. 

EXAMPLES. 

1.  Find  the  sum  of  35.25  meters,  76.45  meters,  89.28 
meters,  and  36.46  meters. 

2.  Find  the  sum  of  75.45  grams,  84.67  grams,  45.84 
grams,  and  97.52  grams. 

3.  Find  the  sum  of  125.65  ares,  223.87  ares,  97.423 
ares,  and  867.055  ares. 

4.  Find  the  sum  of  72.125  liters,  99.72  liters,  125.406 
liters,  and  237.125  liters. 

5.  A  man  bought  at  one  time  26.25  hectoliters  of 
grain,  at  another  time  38.50  hectoliters,  at  another,  46.75 
hectoliters,  and  at  another,  86.25  hectoliters;  how  much 
did  he  buy  in  all  ? 

6.  Mr.  Behmer  bought  at  one  time  26.25  kilos  of 
meat,  at  another  time  38.75  kilos,  at  another,  29.50 
kilos,  and  at  another,  46.75  kilos  j  how  much  did  he  buy 
in  all  ? 

7.  A  man  owns  three  farms ;  the  first  contains  916.25 
ares,  the  second  829.50  ares,  the  third  765.75  ares,  the 
fourth  1227.75  ares ;  how  many  ares  does  he  own  in  all  ? 

8.  A  merchant  sold  6.25  liters  of  molasses  to  one  man, 
12.75  liters  to  another,  8.50  liters  to  another  and  25.25 


THE   METRIC    SYSTEM.  161 

Titers  to  another ;  how  many  liters  of  molasses  did  he 
sell  to  the  four  men  ? 

9.  A  postmaster  mailed  five  letters  this  morning ;  the 
first  weighed  2.25  grams,  the  second  5.50  grams,  the 
third  4.85  grams,  the  fourth  6.65  grams,  and  the  fifth 
7.54  grams ;  what  did  they  all  weigh  ? 

SUBTRACTION   IN   THE    METRIC    SYSTEM. 
184.  The  Metric  System  being  based  upon  the  deci- 
mal system,  subtraction  is  performed   the  same  as  in 
simple  numbers. 

EXAMPLES. 

1.  Subtract  35.75  meters  from  53.20  meters. 

2.  Subtract  96.73  grams  from  104.34  grams. 

3.  Subtract  378.25  ares  from  523.40  ares. 

4.  A  man  bought  72.125  liters  of  wine,  and  sold  36.375 
liters ;  how  much  did  he  retain  ? 

5.  If  I  own  906.25  ares  of  land  and  sell  376.42  ares, 
how  much  will  remain  ? 

6.  If  I  buy  82  hectoliters  of  grain,  and  sell  962  liters 
of  it ;  how  much  will  remain  ? 

7.  If  my  father  owns  1064  ares  of  land,  and  sells  565 
square  meters,  how  much  will  remain  ? 

8.  From  a  barrel  containing  150  liters  of  wine,  I 
drew  out  47.25  liters,  and  there  leaked  out  25.37  liters  j 
how  much  remained  in  ? 

9.  A  man  had  127.45  hectares  of  land ;  he  sold  75  ares 
to  one  man,  and  148  ares  to  another ;  how  many  ares  of 
land  remained  ? 

10.  A  merchant  had  a  rope  one  kilometer  in  length  ; 
he  sold  25  meters  to  one  man,  34  meters  to  another,  and 
128  meters  to  another ;  how  many  meters  of  the  rope 
remained  ? 

11.  A  man  bought  167.5  kilos  of  sugar,  and  sold  83.25 
kilos  j  how  much  sugar  remained  ? 

14* 


162  THE    METRIC    SYSTEM. 

MULTIPLICATION   IN   THE   METRIC   SYSTEM. 

185.  The  Metric  System  being  based  upon  the  deci- 
mal system  of  notation,  multiplication  is  performed  as 
in  simple  numbers. 

EXAMPLES. 

1.  Multiply  28.25  grams  by  5. 

2.  Multiply  35.76  meters  by  8. 

3.  If  a  new  five-cent  piece  weighs  5  grams,  what  will 
25  of  such  pieces  weigh  ? 

4.  If  the  diameter  of  the  five-cent  piece  is  2  centi- 
meters, what  is  the  length  of  a  row  of  50  of  them? 

5.  If  one  decistere  of  wood  is  worth  36  cents,  what 
are  25.5  decisteres  worth  ? 

6.  How  much  will  28.25  liters  of  cider  cost  at  15  cents 
a  liter  ? 

7.  Henry  bought  28 f  hectoliters  of  grain  at  $4.25  a 
liter;  what  did  he  pay  for  it? 

8.  Mr.  Baker  bought  56f  hectoliters  of  grain  at  $3.50 
a  liter,  and  sold  it  at  $3.25  a  liter;  what  did  he  lose  ? 

9.  A  man  travelled  at  the  rate  of  128.5  kilometers  in 
a  day ;  how  far  did  he  travel  in  24  days  ? 

10.  A  laborer  can  disc  4.25  cubic  meters  of  ditch  in  a 
day ;  how  much,  at  that  rate,  can  he  dig  in  5£  days  ? 

DIVISION   IN   THE    METRIC   SYSTEM. 

186.  The  Metric  System  being  based  upon  the  deci- 
mal system  of  notation,  division  is  performed  as  in 
simple  numbers. 

EXAMPLES. 

1.  Divide  17.25  grams  by  5. 

2.  Divide  29.25  meters  by  15. 

3.  If  25  meters  of  cloth  cost  $86.25,  what  will  one 
meter  cost  ? 

4.  If  27  hectares  of  land  cost  $7701.75,  what  will  one 
hectare  cost  ? 


THE    METRIC    SYSTEM.  163 

5.  How  much  will  a  gram  of  jewels  cost,  if  12  grams 
And  5  decigrams  cost  $81.25  ? 

6.  How  many  liters  of  wine  can  you  buy  for  $38.12  £, 
at  the  rate  of  $1.25  a  liter?  Ans.  30.5  liters. 

7.  If  5  dekasteres  of  wood  cost  $12.75,  what  must  I 
pay  for  8  hectosteres,  6  decisteres  of  wood  ? 

8.  What  cost  one  meter  of  cloth,  if  36  meters,  4  deci- 
meters, and  5  centimeters  cost  $169.49  ?       Ans.  $4.65. 

9.  I  paid  $194.18  for  sugar,  at  the  rate  of  56  cents  per 
kilogram ;  how  many  kilograms  did  I  buy  ? 

10.  If  48.625  meters  of  cloth  cost  $183.75,  for  what 
ought  I  to  sell  9.725  meters  so  as  neither  to  gain  nor 
lose  any  thing  ?  Ans.  $36.75. 


MISCELLANEOUS    PROBLEMS. 

1.  The  new  5-cent  piece  weighs  5  grams ;  how  much 
will  50  of  them  weigh  ? 

2.  The  diameter  of  the  new  5-cent  piece  is  2  centi- 
meters ;  how  long  will  a  row  of  50  of  them  be  ? 

Ans.  1  meter. 

3.  A  meter  of  cloth  costs  52  cents ;  what  will  a  hecto- 
meter cost  at  the  same  rate  ? 

4.  One  decistere  of  wood  is  worth  32  cents ;  what  is 
a  stere  of  wood  worth  ? 

5.  If  a  liter  of  wine  weighs  872  grams,  what  will  a 
hectoliter  of  wine  weio-h  ? 

6.  If  a   letter   weighs   2.5   grams,   how   many   such 
letters  will  it  take  to  weigh  a  kilogram  ? 

7.  A  hectoliter  of  corn  cost  $51;   at  this  rate,  what 
will  a  liter  of  corn  cost  ? 

8.  Mary  bought  10.5  meters  of  silk  for  a  dress,  at  $4.50 
a  meter;  what  did  it  cost  her? 

9.  A  grocer  bought  186  kilograms  of  sugar  at  18 } 
eents  a  kilo  ;  what  did  it  cost  ? 

10.  I  bought  5£  kilos  of  beef  at  37£  cents  a  kilo ;  how 
much  did  it  cost  me  ? 


164  THE    METRIC    SYSTEM. 

11.  A  man  sold  36  hectares  of  land  at  $4.25  a  hectare ; 
what  did  he  receive  for  it  ? 

12.  What  must  I  pay  for  324  liters  of  coal  oil,  if  it 
cost  me  15?  cents  a  liter? 

13.  If  a  kilogram  weighs  about  2\  pounds,  how  much 
will  a  myriagram  weigh  ? 

14.  If  a  tonneau  equals  1000  kilos,  how  many  myria- 
grams  in  3  tonneaux?  Ans.  300 M.G. 

15.  If  a  kilogram  of  coffee  cost  96  cents,  what  will 
a  tonneau  of  coffee  cost  ? 

16.  If  a  tonneau  equals  1000  kilos,  how  many  ton- 
neaux in  16700  kilos?  Ans.  16.70  tonneaux. 

17.  A  "quintal  is  1  tenth  of  a  tonneau ;  how  many 
kilograms  are  there  in  a  quintal  ? 

18.  If  a  kilogram  of  pork  costs  37£  cents,  what  will  a 
quintal  of  pork  cost  ? 

19.  If  a  man  walks  6.5  kilometers  in  an  hour,  how 
far  will  he  walk  in  12  hours  ? 

20.  If  an  are  of  land  is  worth  $26.50,  what  are  7 
hectares  of  land  worth  ? 

21.  I  bought  1600  ares  of  land  at  $15  £  an  are,  and 
sold  it  at  $175  a  hectare  j  how  much  did  I  gain  ? 

22.  I  bought  15  kilograms  of  drugs  at  $27.50  a  kilo, 
and  retailed  them  at  the  rate  of  5  cents  a  gram ;  what 
did  I  gain  ? 

23.  A  cask  of  cider  lost  by  leakage  144L.  in  6  hours; 
how  much  leaked  out  in  an  hour  ? 

24.  If  5  steres  of  wood  cost  $7.50,  what  must  I  pay  for 
7.5  steres  of  the  same  kind  of  wood  ? 

25.  If  the  height  of  a  pole  is  54.57GM.,  how  long  will 
it  take  a  worm  to  climb  to  its  top  at  the  rate  of  12 M.  a 

day? 

26.  If  a  kilogram  of  sugar  costs  18:]  cents,  how  many 
kilograms  can  be  bought  for  75  dollars? 

27.  A  man  bought  a  lot  of  land  containing  45  hectares, 
and  sold  from  it  150  square  meters;  how  much  remained? 


THE    METRIC    SYSTEM.  165 

28.  If  6.5  meters  of  cloth  cost  £9.25,  what  will  be  the 
cost  of  32.5  meters  at  the  same  rate  ? 

29.  A  block  of  marble  2.5  meters  long,  .75  meters 
wide,  and  .5  meters  thick  cost  $10 ;  what  would  a  cubic 
meter  of  marble  cost  at  the  same  rate  ? 


EEDUCTION 

FROM    ONE    SYSTEM    TO    THE    OTHER. 

187.  Until  the  Metric  System  has  gone  into  general 
use,  it  will  be  necessary  to  reduce  from  one  system  to 
the  other ;  hence  the  following  exercises  are  presented. 

MEASURES    OF    LENGTH. 

188.  A  3Ieter  equals  39.37in.,  or  3.28ft. ;  hence,  to  re- 
duce from  one  measure  of  length  to  another,  we  have 
the  following  rules  : — 

Eule  I. — Multiply  the  number  of  meters  by  39.37,  and 
it  will  give  the  number  of  inches;  or  by  3.28,  and  it  will  give 
the  number  of  feet. 

Eule  II. — Divide  the  number  of  inches  by  39.37.  or  the 
number  of  feet  by  3.28,  and  it  will  give  the  number  of  meters. 

1.  How  many  inches  in  16  meters  ? 

2.  How  many  feet  in  2.5  meters  ? 

3.  How  many  rods  in  25  meters  ? 

4.  How  many  miles  in  12.5  kilometers  ? 

5.  How  many  meters  in  9.20  feet  ? 

6.  How  many  meters  in  629.92  inches? 

MEASURES    OF    SURFACE. 

189.  An  Are  equals  119.6sq.  yd.,  or  0.00247  acre; 
hence  we  may  readily  reduce  from  one  measure  of  sup 
face  to  the  other. 

1.  How  many  sq.  yd.  in  25  ares  ? 

2.  How  many  ares  in  2990sq.  yd.  ? 

3.  How  many  acres  in  360  ares? 


166  THE    METRIC    SYSTEM. 

4.  How  many  ares  in  8.892  acres  ? 

5.  How  many  ares  in  20A.  2E.  ? 

6.  How  many  hectares  in  20  A.2E.  ? 

MEASURES    OF   VOLUME. 

190.  A  Steve  equals  35.3166  cubic  feet,  or  1.308  cu  yd., 
or  .2759  cord ;  hence  we  may  readily  reduce  from  one 
measure  to  the  other. 

1.  How  many  cu.  ft.  in  25  steres  ? 

2.  How  many  cu.  yd.  in  3.5  steres  ? 

3.  How  many  steres  in  8829.15  cu.  ft.  ? 

4.  How  many  steres  in  4.578  cu.  yd.  ? 

5.  How  many  cords  in  12.5  steres  ? 

6.  How  many  steres  in  34.4875  cords? 

MEASURES    OF    CAPACITY. 

191.  A  Liter  equals  1.0567  quarts  liquid  measure,  or 
.908  quarts  dry  measure;  hence  we  can  readily  reduce 
from  one  measure  to  the  other. 

1.  How  many  liquid  quarts  in  23  liters? 

2.  How  many  dry  quarts  in  50  liters  ? 

3.  How  many  gallons  in  23  liters  ? 

4.  How  many  bushels  in  5  hectoliters  ? 

5.  How  many  liters  in  24.3041  liquid  quarts  ? 

6.  How  many  liters  in  45.40  dry  quarts? 

MEASURES    OF    WEIGHT. 

192.  The  Gram  equals  15.432  Troy  grains;  the  Mo- 
gram  equals  2.2046  lbs.  avoirdupois;  hence  we  can  readily 
reduce  from  one  system  to  the  other. 

1.  How  many  grains  in  4.25  grams? 

2.  How  many  pounds  in  8.5  kilograms? 

3.  How  many  grams  in  65.586  grains? 

4.  How  many  kilograms  in  18.7391  pounds? 

5.  How  many  pounds  in  16  quintals? 

6.  How  many  pounds  in  24  tonneaux  ? 


THE    METRIC    SYSTEM. 


167 


To  assist  pupils  in  becoming  familiar  with  the  value 
of  the  different  denominations  of  the  Metric  System,  we 
present  the  following  Tables. 

193.  Tables  showing  the  Kelation  of  the  De- 
nominations of  the  Metric  System  to  the  Common 
System. 

Measures  of  Length. 


Names. 

Value  in  Meters. 

Value  in  the  Common  System. 

Millimeter, 

.001 

.0394  inch. 

Centimeter, 

.01 

.3937  inch. 

Decimeter, 

.1 

3.937  inches. 

Meter, 

1. 

39.37     inches. 

Dekameter, 

10 

393.7      inches. 

Hectometer, 

100 

328^   feet. 

Kilometer, 

1000 

.62137  mile. 

Myriameter, 

10000 

6.2137  miles. 

Measures  of  Surface. 


Names. 

Value  in  M2. 

■i 
Value  in  the  Common  System. 

Centare, 
Are, 
Hectare, 

1 

100 

10000 

1550   sq.  in. 
119.6  sq.  yd. 
2.471  acres. 

Measures  of  Capacity. 


Names. 

Value 
in  Liters. 

Dry  Measure. 

Wine  Measure. 

Milliliter, 

.001 

.061  cu.  in. 

.27  fluid  dr'm. 

Ten  til  iter, 

.01 

.6102  cu.  in. 

.338  fluid  oz. 

Deciliter, 

.1 

6.1022  cu.  in. 

.845  pill. 

Liter, 

1 

.908  qt. 

1.0567  qt. 

Dekaliter, 

10 

9. OS  qt. 

2  6417  gal. 

Hectoliter, 

100 

2.8375  bu. 

26.417  gal. 

Kiloliter,  or  St  ere, 

1000 

1.308  cu.  yd. 

264.17  gal. 

168 


THE    METRIC    SYSTEM; 


Weights. 


Names. 

Value 
in  Grams. 

Common  Weights. 

1 
Quantity  of  Water. 

Milligram, 

.001 

.0154  gr.  Tr. 

1  m.  m3. 

Centigram, 

.01 

.1543  gr.    " 

10  m.  m3. 

Decigram, 

•.1 

1.543  gr.     " 

T\  c.  m3. 

Gram, 

1 

15.432  gr.      " 

1  c.  m3. 

Dekagram, 

10 

.3527  oz.  av. 

10  c.  m3. 

Hectogram, 

100 

3.5274  oz.    " 

1  deciliter. 

Kilogram, 

1000 

2.2046  lb.    " 

1  liter. 

Myriagram, 

10,000 

22.046  lb.      " 

10  liters. 

Quintal, 

100,0U0 

220.46  lb.         » 

1  hectoliter. 

Tonneau, 

1,000,000 

2204.6  lb. 

1  M3,  or  1  K.  L. 

Common  System  compared  with  the  Metric. 


1  inch 

=    .0254  meter. 

1  gallon    = 

3.786  liters. 

1  foot 

=    .3048  meter. 

1  bushel    = 

.3524  hectoliter. 

1  yard 

==    .9144  meter. 

1  cu.  feet  == 

.2832  hectoliter. 

1  mile 

=  1.6094  kilometer. 

1  cu.  yd.    = 

.7646  stere. 

1  sq.  ft. 

=    .0929  sq.  meter. 

1  cord        = 

3.625  steres. 

1  sq.  yd. 

=    .8362  sq.  meter. 

1  grain      = 

.0648  gram. 

1  sq.  rd. 

=    .2529  are 

1  Av.  oz.   = 

.0283  kilogram. 

1  acre 

=    .4047  hectare. 

1  lb.  Troy= 

.373  kilogram. 

1  sq.  mile 

=    .259  hectare. 

1  lb.  Av.    = 

.4536  kilogram. 

1  quart 

==    .9465  liter. 

1  ton          = 

.9071  tonneau. 

PERCENTAGE.  169 


SECTION   VIII. 

PEBCENTAGE. 

194.  Percentage  is  a  process  of  computation  in  which 
the  basis  of  comparison  is  a  hundred. 

195.  The  term  per  cent,  means  by  or  on  a  hundred; 
thus,  5  per  cent,  of  any  thing  means  5  of  a  hundred  of  it. 

196.  Hence,  1  per  cent,  of  a  number  is  yi^  of  it ;  2 
per  cent,  is  t§q  of  it ;  5  per  cent,  is  T^  of  it,  etc.  It  is 
also  evident  that  100  per  cent,  of  a  number  is  the  whole 

01   it. 

197.  The  sign  of  Percentage  is  %,  and  is  read  per 
cent.  The  per  cent,  is  also  indicated  by  a  common  frac- 
tion or  a  decimal ;  thus,  5%  =  T^  =  .05. 

19S.  In  percentage  there  are  four  quantities  con- 
sidered : 

1.  The  Base,  or  number  on  which  percentage  is 
estimated. 

2.  The  Pate,  or  number  denoting  how  many  of  a 
aaudred. 

3.  The  Percentage,  denoting  how  many  of  the  basis'. 

4.  The  Amount  or  Difference  of  the   basis  and   per- 


centage. 

RATE    EXPRESSED    : 

BY    A 

FRACTION. 

2  per  cent. 

equals 

.02, 

or  Too'  or 

1 
50" 

4  per  cent. 

a 

.04, 

a        4         a 
TTJO' 

1 
25"' 

5  per  cent. 

u 

.05, 

a       5        " 
l  oO' 

1 
20' 

20  per  cent. 

u 

.20, 

a      2  0       a 
TOO' 

1 
5' 

25  per  cent. 

u 

.25, 

U      2  5        it 

i  oO' 

1 
4~* 

125  per  cent. 

it 

1.2&, 

U     125       U 
TOO' 

5 
4* 

A  per  cent,  is  .00.^,  or  .005  ;  i  per  cent,  is  .001,  or  .002; 
12£  per  cent,  is  .12^,  or  .125,  etc. 


15 


0 

PERCENTAGE. 

EXERCISES. 

Express 

decimally 

Express  in  a 

common  fraction 

1. 

6%. 

Ans.  .06. 

6.  6%. 

Ans.  3V 

2. 

12%. 

Ans.  .12. 

7.  12%. 

Ans.  2V 

3. 

163%. 

Ans.  .16?. 

9 

8.  50%. 

Ans.  }2. 

4. 

24%. 

Ans.  .24. 

9.  121%. 

Ans.  I. 

5. 

33|%. 
Express 

Ans.  .33|. 
as  a  per  cent. 

10.  16g%. 

Ans.  J. 

1. 

l 

4* 

Ans.  25  % . 

4     ! 

Ans.  12*  %, 

o 

Li, 

1 

Ans.  20%. 

5.  J. 

Ans.  16f%, 

3. 

1 

2* 

Ans.  50%. 

6.  1. 

Ans.  66f%. 

Case  I. 

199.  Given   the  base  and   rate,  to  find  the  per- 
centage. 

1.  What  is  5  per  cent,  of  $250? 

OPERATION. 

SoLtiYiON.— 5  per  cent,  of  $250  is  Tfo  of  $250,  $250 

or  .05  times  $250,  which,  by  multiplying,  we  find  .05 

to  be  $12.50.     Hence  the  following  «->2  50 

Eule. — Multiply  the  base    by  the  rate,  expressed  deci* 
molly. 


2.  What 

3.  What 

4.  What 

5.  What 

What 
What 
What 
What 
What 
"What 
What 
What 


6 
7 
8 
9 
10 
11 

19 


13 


is  5  per  cent,  of  $280  ? 
is  6  percent,  of  $190? 
is  7  per  cent,  of  $240  ? 
is  8  per  cent,  of  125yds.  ? 
is  9  per  cent,  of  3641bs.? 
is  10  per  cent,  of  982ft.  ? 
is  12  per  cent,  of  831in.? 
is  12^  per  cent,  of  320oz.? 
is  16-  per  cent,  of  630yds.  ? 
is  35  per  cent,  of  1286  miles  ? 
is  40  per  cent,  of  2467  pounds? 
is  75  pei  cent,  of  3182  perches  * 


Ans.  $14. 
Ane.  $11.40. 
Ans.  $16.80. 
Ans   10yds. 


PERCENTAGE.  171 

14.  A  man  bought  a  cow  for  $30,  and  sold  her  at  a 
gain  of  25%  ;  what  did  he  gain?  Ans.  $7h 

15.  A  man  bought  a  horse  for  $150,  and  sold  him  at 
a  gain  of  30%  ;  how  much  did  he  gain  ?         Ans.  845. 

16.  A  lady  bought  360  acres  of  land,  and  sold  12i% 
of  it ;  how  much  did  she  retain  ?  Ans.  315  acres. 

17.  A  man  bought  a  horse  for  $4250,  and  sold  it  at  a 
gain  of  5%  ;  how  much  did  he  receive  for  it  ? 

18.  My  salary  is  $1500  a  year ;  I  spend  25%  of  it  the 
first  half  year,  and  35%  the  second  half;  how  much  do 
I  save  in  a  year  ?  Ans.  $600. 

Case  IT. 

200.  Given  the  percentage  and  rate,  to  find  the 
base. 

1.  60  is  5  per  cent,  of  what  number? 

Solution.— If  5%  of  a  number  is  60,  1  %  operation. 

of  the  number  is  \  of  60,  or  12,  and  100%  5%  =60 

of  the  number,  which  is  the  whole  number,  100%  -—  1200  Ans. 

is  100  times  12,  or  1200.  or, 

Since  this  is  equivalent  to  multiplying  60-f-.05  =  1200 
by  100  and  dividing  by  5,  and  this  last  is 
equivalent  to  dividing  by  .05,  we  have  the  following 

Eule. — Divide  the  percentage  by  the  rate  expressed 
decimally. 

Note. — For  young  pupils  the  analysis  will  be  simpler  than  the  solution 
by  the  rule. 

2.  45  is  20  per  cent,  of  what  number?       Ans.  225. 

3.  75  is  25  per  cent,  of  what  number?        Ans.  300. 

4.  96  is  20  per  cent,  of  what  number? 

5.  230  is  5  per  cent,  of  what  number? 

6.  1121b.  is  40  per  cent,  of  what  weight  ? 

7.  456  acres  are  30  per  cent,  of  how  many? 

8.  237  cows  are  25  per  cent,  of  how  many  ? 

Ans.  948. 
*  9.  157yds.  are  12^  per  cent,  of  how  many  ? 

Ans.  1256, 
13* 


Y12  PERCENTAGE. 

10.  644  pigs  are  35  per  cent,  of  how  many  ? 

11.  $78.18  is  33J  per  cent,  of  how  much? 

Ans.  $234.54. 

12.  A  man  spends  $500  a  year,  which  is  25%  of  his 
salary  ;  what  is  his  salary  ?  Ans.  $2000. 

13.  A  has  280  acres  of  land,  and  this  is  35%  of  what 
B  has ;  how  much  has  B  ? 

14.  A  sold  36  pigs,  which  is  8  %  of  what  he  now  has ; 
how  many  had  he  before  the  sale  ? 

15.  A  boy  found  $15,  which  is  30%  of  what  he  then 
had  ;  how  much  had  he  at  first  ?  Ans.  $35. 

16.  A  man  had  $13681.60  in  a  bank,  and  drew  out 
35  %  of  it ;  how  much  did  he  draw  out  ? 

Ans.  $4788.56. 

Case  III. 

201.  Given  the  base  and  percentage,  to  find  the 
rate. 

1.  25  is  what  per  cent,  of  125  ? 

OPERATION. 

Solution.— 125  is  100  per  cent,  of  itself,  125  =  100% 

and  25,  which  is  ffc  of  125,  is  ffe  of  100  25  =  T2^  X  100% 

per  cent.,  or  \  of  100  per  cent,,  which  is  =$X1(>0%  =20% 
20  per  cent.     Hence  the  following 

Eule. —  Take  such  a  part  of  100  as  the  percentage  is  of 
the  base;  or,  multiply  the  percentage  by  100  and  divide  by 
the  base. 

2.  75  is  what  per  cent,  of  300  ?  Ans.  25%. 

3.  90  is  what  per  cent,  of  360  ?  Ans.  25%. 

4.  45  is  what  per  cent,  of  225  ?  Ans.  20%. 

5.  72  is  what  per  cent,  of  216  ?  Ans.  33£%. 

6.  96  is  what  per  cent,  of  128  ? 

7.  48  is  what  per  cent,  of  120  ? 

8.  112  is  what  per  cent,  of  896  ? 

9.  A  man  had  $960,  and  lost  $240  ;  what  per  cent,  did 
h°>  lose  ? 


PERCENTAGE.  173 

10.  B  lost  825,  and  then  had  $125 ;  what  per  cent,  of 
his  money  did  he  lose  ? 

11.  C  sold  50  cows,  which  was  25  per  cent,  of  the  re- 
mainder; how  many  had  he  at  first? 

12.  D  gave  his  sister  $180,  and  had  $960  left;  his 
money  is  now  what  per  cent,  of  what  he  had  at  first  ? 


SIMPLE  INTEREST. 

202.  Interest  is  money  charged  for  the  use  of  money. 
It  is  estimated  at  a  certain  rate  per  cent,  per  annum. 

203.  The  Principal  is  the  sum  on  which  interest  is 

reckoned. 

204.  The  Rate  of  Interest  is  the  interest  on  100  for 

one  year. 

205.  The  Time  is  how  long  the  money  is  on  interest. 

206.  The  Amount  is  the  sum  of  the  principal  and 

interest. 

207.  Simple  Interest  is  interest  upon  the  principal 
only.  Compound  Interest  is  interest  upon  the  prin- 
cipal and  interest. 

208.  Legal  Interest  is  the  rate  established  by  law. 
Usury  is  a  rate  greater  than  the  legal  rate.  The  taking 
of  usury  is  prohibited  by  law. 

209.  The  quantities  considered  in  Simple  Interest 
are  the  Principal,  Bate,  Time,  Interest,  and  Amount 
There  are  four  cases. 

Case  I. 

210.  Given  the  principal,  rate  per  cent.,  and 
time,  to  lind  the  interest  or  amount. 

First  Method. 
1.  What  is  the  interest  of  $2400,  for  6yr.  7mo.  15da.. 
at7%?  lft. 


174  PERCENTAGE. 

Solution. — By  reduction  we   find  that   6yr.  deration 

Tmo.   15da.  equal  C|yr.     If  the  interest  of  $1  $2400 

for  lyr.  is  7ct.,  the  interest  of  $2400  for  lyr.  is  .07 

2400  times  7ct.,  which  is  $168,  and  for  6§yr.  it  168.00 

is  6|  times  $168,  which  by  multiplying  we  find  g| 

is  $1113.     Hence  the  following  $1113  00  Ans 

Eule. — Multiply  the  principal  by  the  rate  per  cent.t 
expressed  decimally,  and  that  product  by  the  time  expressed 
in  years. 

Kequired  the  interest 

2.  Of  $180  for  3yr.  6rao.  at  7%.  Ans.  $44.10. 

3.  Of  $470  for  7yr.  8mo.  at  6%.  Ans.  $216.20. 

4.  Of  $172  for  5yr.  9mo.  at  5%.  Ans.  $49.45. 
5    Of  $480  for  5yr.  lOmo.  at  12%. 

8.  Of  $1080  for  3yr.  7mo.  6da.  at  5%. 

7.  Of  $1260  for  2yr.  2mo.  12da.  at  5%. 

8.  Of  $1000  for  3yr.  8mo.  12da.  at  10%. 

Second  Method. 

211.  The  second  method,  called  the  "  Six  per  cent, 
method,"  is  perhaps  the  method  most  generally  used  by 
business  men. 

1.  What  is  the  interest  of  $240  for  2yr.  8mo.  12da. 
at  6%  f 

Solution. — 2yr.  8mo.  equal  32mo.    The  in-  operation. 

terest  of  $1  for  12mo.  is  6cts.,  and  for  lmo.  it     2yr.  8mo.  =  32mo. 
is  T\  of   6cts.,   or  Jet,  and  for  32mo.   it  is  32X£  =  $0-16 

32><^  =  16cts.  Also,  since  the  interest  on  $1  12X1-       -002 

for  lmo.,  or  30cla.,  is  Jet.  or  5  mills,  for  Ida.  it  $0,162 

is  3L  of  5  mills,  or  l  of  a  mill,  and  for  12da.  240 

it  is  12  X  \  =  2  mills :    uence  tne  interest  on  $38~88 

#1  for  32mo.  and  12da.  is  16cts.  plus  2  mills, 

or  $0,162.     If  the  interest  on  $1  is  $0,162,  on  $240  ft  is  240  times 
$0,162  which  equal    $38.88.     From  this  we  have  the  following 


PERCENTAGE.  175 

Rule  — 1,  Call  one-half  of  the  number  of  months  tents, 
and  one-sixth  of  the  number  of  days  mills,  and  their  sum 
will  be  the  interest  of  one  dollar,  for  the  given  time,  at 
6  per  cent. 

2.  Multiply  this  by  the  principal,  and  the  product  will  be 
the  entire  interest  at  6  per  cent.  For  any  other  rate,  take 
as  many  sixths  of  it  as  that  rate  is  of  six. 

Notb.— 1.  For  1%  add  J,  for  Sf0  add  J,  for  9%  add  %,  for  5%  subtract 
J,  for  4  %  subtract  J,  etc. 

2.  When  the  time  is  brief,  the  rule  of  business  men  is  as  follows :  "  Multi- 
ply dollars  by  days,  and  divide  by  60." 

Required  the  interest 

1.  Of  $360  for  5yr.  6mo.  12da.  at  6%. 

Ans.  8119.52. 

2.  Of  $480  for  3yr.  8mo.  18da.  at  6%. 

Ans.  $107.04. 

3.  Of  8256  for  7yr.  4mo.  24da,  at  6%. 

Ans.  8113.66. 

4.  Of  848.25  for  3yr.  6mo.  6da.  at  6%. 

Ans.  810.18. 

5.  Of  850.50  for  6yr.  lOmo.  18da.  at  7%. 

Ans.  824.33. 

6.  Of  828.25  for  5yr.  7mo.  24da.  at  5%. 

7.  "What  is  the  amount  of  8360  for  2yrs.  and  6mo.  at 
6  per  cent,  ?  Ans.  8414. 

8.  What  is  the  amount  of  $250  for  Syr.  8mo.  18da. 
at  6  per  cent.  ?  Ans.  8305.75. 

9.  What  is  the  amount  of  $620  for  5yr.  lOmo.  24da. 
at  7  per  cent.  ?  Ans.  8876.06. 

10.  Mary's  father  put  out  8500  on  interest,  at  10  %  at 
her  birth ;  how  much  will  she  be  worth  when  she  is  21 
years  of  age  ? 

11.  Required  the  difference  between  the  interest  and 
the  amount  of  $624  for  3yr.  8mo.  15da.,  at  5  per  cent. 

For  other  exercises  under  this  rule  solve  the  problems  in  the  pre- 
vious and  following  cases. 


176 


PERCENTAGE. 


Third  Method. 

212.  A  metfhocl  of  computing  interest  by  taking 
aliquot  parts. 

1.  What  is  the  interest  of  $2400  for  6yr.  7mo.  15da. 

at  7%? 

Solution.  —  We  find 
the  interest  for  1  yr.  and 
then  for  6yr.,  and  then 
proceed  thus:  7mo.  = 
Gmo.  plus  lmo.,  and 
since  6mo.  equal  \  of  a 
year,  the  interest  for 
6mo.  is  \  of  $1G8,  which 
is  $84 ;  and  since  lmo. 
is  i  of  6mo.,  the  interest 


OPERATION. 

$2400 

.07 

168.00  =  Int.  for  lyr. 

6 

1008.00  =  Int  for6yr. 

6mo. 

=$yr- 

84.00  =  Int.  for  6mo. 

lmo. 

=  - of  lyr. 

14.00  =  Int.  for  lmo. 

15da. 

-Juio. 

7.00  =  Int.  for  15da. 
1113.00,  Ans. 

for    lmo.    is    \    of    $84, 

which  is  $14 ;   and  since 

15  days  is  £  of  a  month,  the  interest  for  15  days  is  |  of  $14,  which 

is  $7 ;  and  the  whole  interest  is  the  sum  of  these,  which  we  find  to 

be  $1113.     Hence  the  following 

Eule. — 1.  Find  the  interest  for  the  number  of  years,  as 
by  the  first  method. 

2.  Find  the  interest  for  the  number  of  months  by  taking 
convenient  fractional  parts  of  one  year's  interest. 

8.  Find  the  interest  for  the  number  of  days  by  taking 
fractional  parts  of  one  month'' s  interest. 

Required  the  interest 

2.  Of  $780  for  4yr.  8mo.  at  6%.  Ans.  $218.40. 

3.  Of  $960  for  7yr.  9mo.  at  7%.  Ans.  $520.80. 

4.  Of  $1260  for  3yr.  6mo.  15da.  at  8  %.     Ans.  $357. 

5.  Of  $2480  for  5yr.  5mo.  lOda.  at  5%. 

Ans.  $675.11. 


Note. — Let  the  pupil  solve  by  this  method  the  problems  under  the  first 
u.nd  second  methods. 


STOCKS   AND    DIVIDENDS.  177 


STOCKS   AND    DIVIDENDS. 

213.  The  Stoclc  of  a  company  represents  the  money 
mvcsted  in  its  business  by  the  stockholders  or  owners. 

Stock  is  divided  into  equal  parts,  called  shares, 

214.  A  Dividend  is  a  sum  to  be  paid  to  the  stock- 
holders out  of  the  gains  of  the  company.  It  is  divided 
according  to  the  par  value  of  stock  held  by  them. 

215.  The  Par  Value  of  stock  is  its  nominal  value  as 
fixed  by  the  charter,  or  articles  of  agreement,  of  the 
company.  It  is  usually  $50  or  $100  per  share;  although 
other  sums  are  often  agreed  upon. 

216.  The  Heal  Value  of  stock  is  what  it  will  sell  for. 

217.  Premium  is  how  much  its  real  value  exceeds  its 
par  value. 

21S.  Discount  is  how  much  its  real  value  is  less  than 
its  par  value. 

Case  I. 

219.  Given  the  stock  and  rate  of  dividend,  to 
find  tlie  dividend. 

1.  A  owns  50  shares  of  bank-stock,  at  $100  each; 
the  bank  declares  a  dividend  of  6  %  ;  required  A's  divi- 
dend. 

OPERATION. 

Solution. — If  one  share   is  worth  SI 00,  50  50 

shares  are  worth  50  times  $100,  or  $5000.    Hence  100 

Ars  stock  is  worth  $5000.     His  dividend  is  .06  5000 

times  $5000,  or  $300.  .06 

$  30()! 00 

Eule. — Multiply  the  par  value  of  the  number  of  shares 
by  the  rate  of  dividend. 

2.  A  man  owns  56  shares  of  stock,  at  $50  per  shar<? 


178  STOCKS    AND    DIVIDENDS. 

par  value ;  the  company  declares  a  5%  dividend;  what 
is  his  dividend  ?  Ans.  $140. 

3.  A  lady  has  175  shares  of  stock,  at  $10  par  per  share ; 
the  company  declares  8J%  dividend;  what  is  her  divi- 
dend ?  Ans.  $148.75. 

4.  I  hold  250  shares  of  mining  stock,  at  $20  par  a 
share;  the  company  has  divided  h\°J0\  what  is  my 
dividend  ?  Ans.  $275. 

Case  II. 

220.  Given  the  par  value  and  rate  of  premium 
or  discount,  to  find  the  premium,  discount,  or  real 
value. 

1.  A  bought  25  shares  of  stock  ($50),  at  4  %  premium  ; 
required  the  premium  and  the  real  value. 

OPERATION. 

Solution. — The  par  value  of  25  $50  X  25  =  $1250  =  par  value, 
shares  at  $50  a  share  is  25  times  .04 

$50,  or  $1250.     The  premium  is  .04  50.00  =  premium, 

times  $1250,  or  $50,  and  the  pre-  1250. 

mium  added  to  the  par  value  equals  $1300       =  real  value. 

$1300,  the  real  value. 

Eule. — Multiply  the  par  value  by  the  rate,  for  amount  of 
premium  or  discount.  Add  or  subtract  this  from  par  value, 
for  real  value. 

2.  I  bought  75  shares  of  gas  stock  ($20),  at  3$f0  pre- 
mium ;  required  the  premium  and  the  cost. 

Ans.  $52.50;  $1552.50. 

3.  I  sold  120  shares  of  stock  ($15),  at  5%  discount; 
required  the  discount,  and  the  amount  received  for  it. 

Ans.  $90 ;  $1710. 

4.  Sold  34  shares  of  E.  E.  stock  ($50),  at  2%%  dis- 
count; what  was  received  for  it  ?  Ans.  $1657.50. 

5.  Bought  12  shares  bank  stock  ($50),  at  12£%  pre- 
mium, and  afterwards  sold  the  same  at  $60  per  share ; 
did  I  gain  or  lose,  and  how  much  ?     Ans.  Grained  $45. 


STOCKS   AND    DIVIDENDS.  179 

UNITED   STATES    BONDS. 

221.  United  States  Bonds  are  printed  U.S.  promises 
to  pay  the  holder  a  certain  sum  of  money,  with  interest 
payable  at  stated  periods  in  the  interval. 

222.  U.S.  Bonds  may  be  of  various  kinds  : — of  these, 
Seven- Thirties,  Five- Twenties,  Ten-Forties,  and  Twenty- 
years-Sixes,  are  at  present  the  principal.  These  may 
also  be  either  Coujyon  or  Registered  bonds. 

223.  Seven- Thirties  are  so  called  because  they  draw 
an  annual  interest  of  7T30  per  cent.,  or  a  daily  interest 
of  two  cents  on  every  $100.  The  interest  for  less  than 
6  months  is  always  calculated  by  days.  They  are  indi- 
cated thus :  7-30's. 

224.  Five- Twenties  are  so  called  because  they  are 
payable  at  the  option  of  the  Government  any  time 
between  5  years  and  20  years  after  they  are  issued. 
They  are  marked  5-20's. 

225.  Ten-Forties  are  so  called  because  they  are  pay- 
able at  the  option  of  the  Government  any  time  between 
10  years  and  40  years  after  they  are  issued.  They  are 
marked  10-40's. 

226.  Seven-Thirties  as  now  issued  pay  7T30%  interest 
in  legal  currency.  Five-Twenties  pay  6%  interest  in  gold  ; 
and  Ten-Forties,  5%  interest  in  gold.  Interest  on  each 
is  due  semi-annually. 

227.  Coupon  Bonds  have  attached  to  them  "cou- 
pons," or  certificates  for  each  six  months'  interest 
accruing,  payable  as  they  become  due.  These  coupons 
may  be  cut  off,  and  are  payable  to  the  bearer  of  them. 

228.  Registered  Bonds  have  no  coupons  attached, 
and  the  interest  is  payable  only  to  the  bond-holder 
whose  name  is  registered  in  the  Treasury  Department, 
or  to  his  order. 

229.  United  States  bonds  usually  sell  at  a  premium 
over  their  face  value,  and  are  quoted  or  priced  at  their 
current  value  per  S100. 


180  STOCKS   AND    DIVIDENDS. 

Case  I. 

230.  To  find  the  interest  due  on  a  United  States 
Bond. 

1.  What  is  the  semi-annual  interest  of  a  $500  7-30 
bond? 

OPERATION. 

Solution. — If  on  $1  the  interest  for  a  year  is  500 

7T3Q  cents,  on  §500  the  interest  is  500  times  7T3^,  .073 

which  is  $36.50.     If  the  interest  for  1  year  is  2)36  500 

$36.50,  for  £  year  it  is  *  of  $36.50,  or  $18.25.  "       ' 

2.  What  is  the  semi-annual  interest  on  a  $500  5-20 
bond  worth  in  currency,  gold  being  at  a  premium  of 

30%? 

OPERATION. 

Solution. — If  the  interest  for  1  year  is  6%,  500 

for   J   year  it  will   be   3%.       The  interest  on  03 

$500  at  2>f0  is  $15.00.     Since  gold  is  at  a  pre-  15  qq 

mium,  we  must  calculate  the  premium  on  $15  oq 

and  add  it  to  the  $15.     30%  of  $15  is  $4.50,  4  ,-   0Q- 

which   added  to  $15  equals  $19.50,  or  worth  1(- 

of  the  interest  in  currency.  — 

$19.50 

Eule. — 1.  Find  the  interest  on  the  face  of  the  bond  for 
the  given  time  and  rate. 

2.  In  gold-bearing  bonds,  add  the  premium  on  the  interest 
to  the  interest. 

3.  I  have  $4000  7-30  bonds  with  the  half-year's  interest 
due  ;  how  much  is  due  on  them  ?  Ans.  $146. 

4.  My  father  has  a  $5000  5-20  bond  with  a  half-year's 
interest  due ;  how  much  is  the  interest  worth  in  cur- 
rency, gold  at  35%  premium?  Ans.  $202.50. 

5.  I  hold  two  $5000  5-20  bonds  with  the  half-year's 
interest  due ;  how  much  will  I  get  in  "  greenbacks," 
gold  at  25%  premium?  Ans.  $375. 

6.  Miss  Smith  has  $4000  7-30's  and  $4000  5-20's;  for 
which  does  she  get  the  most  interest  in  currency  in  a 
year, — gold  at  35  %  premium  ?     Ans  $32  on  the  5-20's. 


STOCKS    AND    DIVIDENDS.  181 

Case  II. 

231.  To  find  the  cost  of  bonds  when  they  are  at 
a  premium. 

1.  What  must  I  pay  for  a  $500  7-30  bond,  when  7-30'm 
are  at  a  premium  of  5%,  or  sell  at  105  ? 

OPERATION. 

Solution.— Five   per  cent,  of  $500  is  $25,  500 

which   added   to  $500    equals    $525.     Hence  I  .05 

must  pay  $525  for  a  $500  bond  at  5$,  premium.  25.00 

Or,  if  a  $100  bond  is  worth  $105,  my  $500  bond  "ivv^Tno 

is  worth  five  times  as  much,  or  $525.  „,.„. 

'  or  $10o 

5 

$525 

Rule. — 1.  Find  the  premium,  and  add  it  to  the  face  of 
the  bond. 

Eule. — 2.  Multiply  the  price  per  $100  by  the  number  of 
hundreds  bought  or  sold. 

Note. — If  there  is  interest  due  on  the  bond,  it  must  be  added.  If 
a  coupon  not  due  is  cut  off,  deduct  the  unaccrued  interest  thus 
retained. 

2.  What  must  I  pay  for  $2000  7-30's,  when  they  com- 
mand a  premium  of  4£%,  or  sell  at  104}  ?     Ans.  $2090. 

3.  How  much  will  a  $1000  7-30  bond  cost,  when  there 
is  93  days'  interest  due  and  the  premium  is  4^  %  ? 

Ans.  $1066.10. 

4.  When  7-30's  are  at  a  premium  of  5£%,  how  much 
must  I  pay  for  $3500  bonds  with  124  days'  interest  due 
on  them  ?  Ans.  $3779.30. 

5.  I  bought  $3000  5-20's  with  2  months'  interest  due, 
the  bond  being  at  a  premium  of  12%  ;  what  did  it  cost 
me?  Ans.  $3390. 

Note. — No  premium  for  gold  is  allowed  on  less  than  6  months' 
interest. 


16 


182  MISCELLANEOUS    PROBLEMS. 

MISCELLANEOUS   PROBLEMS. 

1.  A  has  $4685,  B  has  $1245  more  than  A,  and  C  has 
as  much  as  A  and  B  both ;  how  many  dollars  has  C  ? 

2.  C  has  438  acres  of  land,  D  has  179  acres  less,  and 
E  has  48  less  than  C  and  D  together ;  how  many  acres 
have  D  and  C  ? 

3.  Henry  can  walk  30  miles  a  day,  and  William  can 
walk  37  miles ;  how  much  farther  can  William  walk  in 
45  days  than  Henry? 

4.  Two  men  start  from  the  same  point  and  walk  in 
opposite  directions,  one  traveling  25,  the  other  32,  miles 
an  hour ;  how  far  apart  will  they  be  in  148  hours  ? 

5.  If  a  boy  can  earn  $28  a  month,  and  a  man  $47  a 
month,  how  much  will  6  boys  and  9  men  earn  in  a 
month  ? 

6.  If  a  clerk  earns  $150  a  month,  and  spends  $48,  how 
much  can  he  save  in  12  months,  or  a  year? 

7.  A  merchant  gave  $8.25  a  barrel  for  96  barrels  of 
flour,  and  sold  it  for  $1000 ;  what  was  the  gain  ? 

8.  How  many  bushels  of  apples  can  you  buy,  at  $2£ 
a  bushel,  for  56  barrels  of  flour,  at  $7.50  a  barrel  ? 

Ans.  168. 

9.  A  farmer  has  137  hens;  now,  if  he  should  lay  out 
$625  for  hens,  at  the  rate  of  25  cents  apiece,  how  many 
would  he  then  have?  Ans.  2637. 

10.  If  a  steamboat  goes  14  miles  an  hour,  and  a  rail- 
road train  32  miles  an  hour,  how  far  will  the  steamboat 
go  while  the  train  goes  672  miles  ?         Ans.  294  miles. 

11.  Mary  and  Susan  had  each  1420  cents;  after  Mary 
had  given  Susan  360  and  Susan  had  given  Mary  480, 
bow  many  had  each  ?     Ans.  Mary,  1540 ;  Susan,  1300. 

12.  Six  men  and  8  boys  earned  a  sum  of  money,  and 
divided  it  so  that  each  man  had  $75  and  each  boy  $63 ; 
bow  much  did  each  earn  ? 

13.  A  had  369  acres  of  land,  then  bought  720  acres, 


MISCELLANEOUS    PROBLEMS.  183 

and  then  divided  the  whole  into  9  equal  farms,  and  sold 
6  of  them  ;  how  many  acres  remain  ? 

14.  A  boy  earns  $2.50  a  day,  and  pays  75  cents  a  day 
for  his  board  j  how  much  can  he  thus  save  in  a  week  ? 

Ans.  $9.75. 

15.  What  number  must  I  add  to  the  product  of  126 
and  72,  to  make  10000? 

1G.  A  lady  went  to  the  city  with  600  eggs,  and  sold 
them  at  15  cents  a  dozen ;  what  did  she  receive  for 
them  ? 

17.  The  sum  of  two  numbers  is  7809,  and  one  of  the 
numbers  is  3725 ;  required  the  other  number,  and  their 
difference. 

18.  The  sum  of  three  numbers  is  4082;  the  first  num- 
ber is  1028,  the  second  2372  ;  required  the  third  number. 

19.  The  difference  between  two  numbers  is  709,  and 
one-half  of  the  smaller  number  equals  482 ;  what  is  the 
larger  number  ? 

20.  A  merchant  bought  26  bales  of  cloth,  each  bale 
containing  32  pieces,  and  each  piece  24  yards;  how 
many  yards  did  he  buy  ? 

21.  If  a  boat  sail  378  miles  in  9  days,  how  far  can  it 
sail  in  15  days  at  the  same  rate  ?  Ans.  630  miles. 

22.  If  28  men  can  build  a  lot  of  wall  in  18  days,  how 
many  men  can  build  the  same  wall  in  21  days  ? 

Ans.  24  men. 

23.  A  drover  bought  365  cows,  at  $25  a  head,  and 
758  sheep,  at  $3.50  a  head  ;  what  was  the  cost  of  all  ? 

24.  A  merchant  bought  96  barrels  of  flour  for  S960 ; 
he  sold  58  barrels  at  $8  a  barrel,  and  the  remainder  at 
$12  a  barrel ;  what  was  the  loss  ? 

25.  What  is  the  sum  of  J,  j,  J,  and  J  ? 

26.  Subtract  the  sum  of  ~  and  f  from  the  sum  of  ^ 
and  J. 

27.  Multiply  f  by  },  and  add  the  result  to  the  product 
of  4  and  j-?-. 


184  MISCELLANEOUS    PROBLEMS. 


28.  Multiply  §  by  |,  and  divide  the  result  by  the  pro- 
ct  of  |  and  ]~. 

29.  Find  the  difference  between  1  off  of  3  and  i  of  js 


off 


30.  If  J  of  a  barrel  of  flour  costs  $5.60,  what  will  4 
barrels  cost  at  the  same  rate  ?  Ans.  $33.60 

31.  If  |  of  a  ton  of  coal  is  worth  $7.50,  what  will  12 
tons  cost  at  the  same  rate?  Ans.  $120. 

32.  A  man  sold  I  of  his  land  for  $9750 ;  what  was  it 
all  worth  at  the  same  rate  ?  Ans.  $16250. 

33.  J\!ary  lost  \  of  her  money,  and  then  had  $960  re- 
maining: how  much  had  she  before  the  loss  ? 

34.  If  §  of  a  ton  of  coal  is  worth  $4.50,  what  is  %  of  a 
ton  worth  at  the  same  rate  ?  Ans.  $9.37^. 

35.  If  |  of  a  lot  of  land  is  worth  $234,  what  is  |  of 
the  same  lot  worth  ?  Ans.  $260. 

36.  When  rye  is  worth  |  of  a  dollar  a  bushel,  how 
many  bushels  can  be  bought  for  J  of  a  dollar  ?      Ans.  J. 

37.  If  one  yard  of  cloth  is  worth  2}  dollars,  how  many 
yards  can  be  bought  for  $5|  ?  Ans.  2T3Qyds. 

38.  A  man  owned  g  of  a  farm,  and  sold  -J  of  his  share ; 
what  part  of  the  whole  farm  remained  ?  Ans.  TV 

39.  Plow  many  times  will  14  gallons  fill  a  vessel  that 
holds  2 1  gallons  ?  Ans.  6  times. 

40.  A  father  divided  510  acres  of  land  equally  among 
his  sons,  giving  them  63|  acres  apiece;  how  many  sons 
had  he  ? 

41.  A  merchant  bought  6|  cords  of  wood,  at  $6§  a 
cord;  and  paid  for  it  with  corn,  at  -|  of  a  dollar  a  bushel; 
how  many  bushels  did  it  take  ?  Ans.  55bu. 

42.  If  the  sum  of  two  fractions  is  |,  and  one  of  them 
is  ~,  what  is  the  other?  Ans.  |-J. 

43.  If  the  difference  of  two  fractions  is  T3y,  and  tl*. 
smaller  is  T\,  what  is  the  other  fraction  ?         Ans.  |J. 

44.  What  fraction  multiplied  by  2|  will  give  a  pru 
duct  of  1J?  Ans.fi. 


MISCELLANEOUS    PROBLEMS.  18l 

45.  A  pole  stands  J  in  the  mud,  |  in  the  water,  and 
20  feet  out  of  the  water;  what  is  the  length  of  the  pole  1 

Ans.  48  feet. 

46.  Reduce  £12  13s.  lOd.  to  pence. 

47.  Reduce  £17  17s.  9d.  3far.  to  farthings. 

48.  Reduce  281b.  lOoz.  16pwt.  22grs.  to  grains. 

49.  Reduce  52876  farthings  to  pounds. 

50.  Reduce  89726  grains,  Troy,  to  pounds. 

51.  Reduce  89726  grains,  Apothecaries',  to  pounds. 

52.  Reduce  31207  drams  to  higher  denominations. 

53.  How  many  drams  in  1  ton  ? 

54.  How  many  inches  in  1  mile  ? 

55.  How  many  square  inches  in  1  acre? 

56.  How  many  square  feet  in  1  square  mile  ? 

57.  How  many  cubic  inches  in  1  cord  of  wood  ? 

58.  How  many  pounds  of  medicine  would  a  physician 
use  in  one  year,  if  he  averaged  10  prescriptions  a  day, 
of  10  grains  each?  Ans.  61b  43  1^. 

59.  A  merchant  bought  6cwt.  321b.  of  sugar  at  8£ 
cents  a  pound  ;  what  did  it  cost  ? 

60.  How  often  will  a  wheel  18ft.  4in.  in  circumference 
revolve  in  going  50  miles?  Ans.  14400  times. 

61.  How  many  square  rods  in  a  rectangular  field  50 
rods  long  and  30  rods  wide  ?  Ans.  1500sq.  rd. 

62.  How  many  acres  in  a  rectangular  field  50  rods 
long  and  30  rods  wide?  Ans.  9A.  1R.  20P. 

63.  How  many  rods  of  fence  will  enclose  the  field  in 
the  preceding  problem  ?  Ans.  160  rods. 

64.  In  an  orchard,  i  of  the  trees  bear  apples,  \  bear 
pears,  and  the  remainder,  which  is  33,  bear  peaches; 
how  many  trees  are  there  in  the  orchard  ? 

Ans.  60  trees. 

65.  How  many  yards  of  carpeting  1  yard  in  width 
will  carpet  a  floor  16  foot  long  and  9  feet  wide  ? 

Ans.  16  yards. 

16* 


IStf  MISCELLANEOUS    PROBLEMS. 

66.  How  many  cubic  feet  in  a  pile  of  wood  25  feet 
long,  12  feet  high,  and  4  feet  wide  ? 

67.  How  many  cords  in  a  pile  of  wood  64  feet  long. 
16  feet  hio-h,  and  4  feet  wide  ?  Ans.  32  cords. 

68.  If  a  load  of  wood  be  16  feet  long  and  6  feet  wide, 
how  high  must  it  bo  to  make  a  cord  ?  Ans.  ljffc. 

69.  If  my  bedroom  is  12  feet  long,  10  feet  wide,  and 
8  feet  high,  and  I  breathe  12  cubic  feet  of  air  in  a  minute, 
how  lone:  will  it  take  to  breathe  as  much  air  as  the 
room  holds?  Ans.  80min. 

70.  America  was  discovered  by  Columbus,  Oct.  11, 
1492  ;  how  long  from  that  time  until  Aug.  8,  1865  ? 

71.  The  Kevolution  commenced  April  19,  1775,  and 
peace  was  declared  January  20, 1783 ;  how  long  did  the 
war  continue  ? 

72.  A  boy  lost  £  of  his  kite-string,  and  then  added  20 
feet,  and  then  found  the  string  was  f  of  its  original 
length ;  what  was  the  length  at  first  ?  Ans.  240ft. 

73.  Mary  spent  20  per  cent,  of  $500  for  a  watch,  and 
20  per  cent,  of  the  remainder  for  a  chain;  how  much 
then  remained  ?  Ans.  $320. 

74.  A  man  had  250  acres  of  land,  and  sold  25%  to  A, 
and  20%  to  B;  how  much  remained? 

Ans.  137£  acres. 

75.  A  man  bought  a  horse  for  $250,  and  sold  it  at  a 
gain  of  20%  ;  what  did  he  receive  for  it?     Ans.  $300. 

76.  A  house  was  bought  for  $1280,  and  sold  for  $1600; 
what  was  the  gain  per  cent.  ?  Ans.  25%. 

77.  Eequired  the  interest  of  $2400  for  6yr.  lOmo. 
24da.,  at  6  per  cent.  Ans.  $993.60. 

78.  Eequired  the  amount  of  $960  for  2yr.  6mo.  12da., 
at  7  per  cent.  Ans.  $1130.24. 

79.  I  would  now  like  each  little  boy  and  girl,  who  has 
gone  through  the  book,  to  tell  me  how  many  months  old, 
then  how  many  weeks  old,  and  then  how  many  days  old, 
he  or  she  is. 

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c » ; 

THE  NORMAL  SERIES 

OF 


ritfcmttitt  m&  IJ^tamita, 


BY 


EDWARD  BROOKS,  A.  M. 


PROFESSOR  OF  MATHEMATICS  IN    THE   PEKNA.  STATE    NORMAL  SCHOOL 

AT  MILLERSVILLE,  PA. 

♦♦..♦♦ • 


BROOKS'S  NORMAL  PRIMARY  ARITHMETIC. 

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BROOKS'S  NORMAL  MENTAL  ARITHMETIC. 

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And  Key  to  Normal  "Written  Arithmetic.  Containing,  in  addition  to 
Solutions  to  Problems,  many  valuable  suggestions  to  Teachers. 

BROOKS'S  NORMAL  GEOMETRY  AND  TRIGONOMETRY. 

Just  Published. 

Designed  as  an  Elementary  Work  for  Popular  Use.  Containing  many 
new  and  simplified  demonstrations. 


OFFICE  OF  CONTROLLERS  OF  PUBLIC  SCHOOLS. 

Piiila.,  March  31,  1864. 
Resolved:  That  Brooks's  Mental  Arithmetic,  Primary  Arithmetic  and  Key,  be  used  in  the  Public 
Schools  of  this  District. JAS.  D.  CAMPBELL,  Secy. 

"His  Mental  Arithmetics  I  consider  the  best  published." 

JOS.  W.  WILSON.  Prof,  of  Pract.  Math., 
Phila.  Central  High  School. 

"  To  those  who  wish  to  carry  out  the  Normal  System  of  teaching  I  can  recommend  no  better  work." 

HENRY  B.  PIERCE,  Prof,  of  Math., 
State  Normal  School,  Trenton,  AT.  J. 

"I  have  no  hesitation  in  saying  that  Prof.  Brooks's  Series  of  Arithmetics  is  the  best  now  in  u-e. 
They  are  more  progressive  than  any  1  have  seen.  I  have  had  better  success  in  teaching  Arithmetic 
with  Brooks's  Analysis  than  any  other.  My  conclusions  are  drawn  from  a  thorough  examination  and 
careful  comparison  of  his  books  with  others."  J.  MORROW,  Teacher  Peebles  District, 

Pittsburg,  Pa. 


PUBLISHED  BY 

SOWER,    POTTS    8x    (DO. 
530  Market  St.  and  523  Minor  St.,  Philada. 


